amenability and weak amenability of banach …
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AMENABILITY AND WEAK AMENABILITY OF BANACH ALGEBRAS
BY
YONG ZHANG
A Thesis Submitted to the Faculty of Graduate Studies
In Partial Fulfillment of the Requirements For the Degree of
Doctor of Philosophy
Department of Mathematics University of Manitoba
Winnipeg, Manitoba
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COPYRIGHT PERMISSION PAGE
AMENABILITY AND 'WEAK AMENABILITY OF BANACH ALGEBRAS
YONG ZEANG
A ThesidPracticnm sabmitted to the Facdty of Graduate Studies of The University
of Manitoba in partial m e n t of the reqoirements of the degree
of
Doctor of Philosophy
Yong Zhang01999
Permission bas been granted to the Lîbrary of The University of Manitoba to lend or seIl copies of this theWpracticum, to the Nationai Libmry of Canada to microfilm this thesis and to lend or seil copies of the film, and to Dissertatioar Abstracts International to pubhh an abstract of this thesis/practicnm.
The author reserves other publication rights, and neither this thesidpracticnm nor extensive extracts from it may be printed or otherwise reprodaced without the author's written permission.
1 would like to express my deepest gratitude to my supervisor, Professor
F. Ghahramani, who was always there for me - with his guidance, caring,
support and encouragement. His extensive mathematical knowledge and
research insight benefited me greatly.
1 wish to thank Professors W. G. Bade, H. G. Dales, N. Gronbaek and
G. Willis for their interest in my work and for their inspiration.
I would also like to thank Professors P. McClure and J. Williams for
carefùlly reading the thesis and for their valuable advice.
My special thanks go to my dear wife Hong Wang. Her understanding
confirmed in me my decision to pursue Mathematics.
My special thanks also go to my daughter Hannah Zhang, whose smiles
always brought me out of iny fiutration and exhaustion.
Finally, I would like to extend my thanks to the University of Manitoba
and the Department of Mathematics for their generous financial support.
We start with the investigation of (2m)-th conjugate algebras of Banach algebras (equipped with Arens products) and related higher order dual Banach modules. In the process we explore important properties of the higher order dual operators of module morphisms and of derivations. Afier this basic investigation we consider various problems in or related to the theory of amenability and weak amenability ofBanach algebras. The work is described by the following three themes:
Theme 1. Weak Amenability of Banach Algebras
We flrst discuss the conditions under which n-weak amenability implies (n+2)-weak amenability for n>O, and the conditions for theunitization of a Banach algebra to be n-weakly amenable. Afier an extensive study of module extensions of Banach algebras, we give a necessary and sufficient condition for a Banach algebra of this kind to be n-weakly amenable, separately for odd numbers n and even numbers n. Then we discuss several concrete cases, especially those related to operator algebras. These finally offer us a way to constmct a counter-example to the open question of whether weak amenability irnplies 3-weak arnenability.
Theme 2. Structure of Contractible Banach Algebras and Amenable Banach Algebras with the Underlying Spaces Refiexive
By exploiting properties of minimal idempotents we show that a Banach algebra of this kind is finite dimensional if every maximal ideal of it is contained in a maximal left ideal which is complemented as a subspace. This result improves several h o w n results concerning the subject.
Theme3. Nilpotent Ideals in a Class of Banach Algebras
We introduce the concept of approximately complemented subspaces in a Banach space. Then we discuss the relation between this notion and some well-known ones like approximation property and weakcomplementability. The category of this notion includes many interesting cases. Tensor products of subspaces of this kind have some natural properties. We prove that in a class of Banach algebras called approximately biprojective Banach algebras, nilpotent ideals, if any, lack certain geometric properties such as being approximately complemented.
Contents
Chapter 1. Introduction
1.1. Preliminaries
1.2. Outline of the content
Chapter 2. Dual Modules
2.1. Arens products
2.2. Bimodule actions of on x ( * ~ )
Chapter 3. The n-Weak Amenability of Banach Algebras
3.1. (n + 2)-Weak amenability and n-weak amenability
3.2- The n-weak amenability of a*
Chapter 4. The n-Weak Amenability of Module Extensions of Banach
Algebras 28
4.1. Lifting derivations 30
4.2. On n-weak amenability of U @ X 40
4.3. The algebra 24 @ % 52
4.4. The algebra U @ X o 61
Chapter 5. Weak Amenability does not imply 3-Weak Amenability 69
5.1. Weak amenability of the algebra 11 8 (XI+&) 1
CONTEhTTS
5.2. Weak amenability is not 3-weak amenability
Chapter 6 . Contractible and Reflexive Amenable Banach Algebras
6.1. Approximation property and contractible Banach algebras
6.2. Maximal ideals and dimensions
Chapter 7. Approximate Complementability and Nilpotent Ideals
7.1. Approximately Complemented Subspaces
7.2. Approximately Biprojective Banach Algebras
7.3. Nilpotent ideals
CHAPTER 1
Introduction
Al1 Banach spaces and algebras we deal with in this thesis are assumed
to be over C, the complex field. Suppose that P is a Banach algebra. A
Banach space X is said to be a Banach left %-module if a bilinear mapping:
(a, x) H ax (called the left module multiplication) of Ill x X into X is defined
which satisfies, for some number k > 0, the following axioms.
Banach right %-modules are defined similarly. X is said to be a Banach
0-bimodule if it is both a Banach left U-module and a Banach right %-module
and the module multiplications are related by
(iii) a(xb) = (ax )b , a, b E M, x E X;
Sometimes ive use a x instead of as to denote the module multiplication to
avoid any possible confusion with other products involved. For any Banach
algebra U, U itself is a Banach a-bimodule with the product of '2 giving the
module multiplications. If X is a Banach left (right) U-module, then X*,
the conjugate space of X, is a Banach right (resp. left) %-module with the
1
1.1. PRELIMrnARIES
natural module multiplications defined by
for f E X', a E U and x E X. We cal1 this module the dual right (resp.
left) module of X. Here and throughout this thesis, for x E X and f E X*,
(x, f ) denotes the value f (x). If X is a Banach 5%-bimodule, then multipli-
cations given by Equation(l.1) make X* into a Banach Il-birnodule, called
the dual module of X. The dual (left, right) module of X' is called the
second dual (resp. le&, right) module of X. In this thesis we use o(X*, X)
to denote the weak* topology on the conjugate space X*. A lirnit under this
topology is called weak* limit, briefly, wk*-lim. Suppose that q5 E X", Then
by a theorem of Goldstine [23, Theorem V.4.51, there exists a net (x,) c X
such that wk*-Iimx, = 4. Then for each a E U it is easy to see that
The nth dual (left, right) module of X can be defined by induction; it will
always be denoted by ~ ( " 1 .
Suppose that X is a Banach left (right) <?L-module. A net (ei) c O is
called a left (resp. right) approxirnate identity for X if limeix = x (resp.
limzei = x) in norrn for al1 x E X. If in addition, (ei) is also a bounded
net, then (ei) is a le& (resp. right) bounded approximate identity, briefly
denoted by left (resp. right) b.a.i.. If X is a Banach U-bimodule and (ei)
is both a left and a right (bounded) approximate identity for X, then (ei)
1.1. PRELIMINAEUES 3
is called a (bounded) approzimate identity fcr X . When X = U, a (left,
right, bounded) approximate identity for X will be called a (resp. left, right,
bounded) approximate identity of U.
The following factorization theorem due to P. J. Cohen (see (4, Theo-
rein 11-10] or [43. Theorern 32-22]) will be often used:
THEOREM 1.1. Suppose that X is a Banach lej? %-module and 2i has a
left b.a . i . for X. Then for any x E X and 6 > O there exist a E II, and
y E X such that x = a y and [lx - y11 5 6.
A special kind of operator which plays an important role in this thesis is
a derivat ion, defined below:
DEFINITION 1.2. Suppose that 2l is a Banach algebra and X is a Banach
Il-bimodule. A (continuous) derivation from % into X is a (continuous)
linear mapping D: U -t X which satisfies
For any x E X, the mapping 6,: U -t X given by
&(a) = a x - xa, (a E I()
is a continuous derivation, called an inner derivation. Denote by Z1(U7 X)
the space of all continuous derivations from % into X and by p ( U , X) the
space of al1 inner derivations from fU into X. Then P(2€, X) is a subspace
1.1. PRELIMINARIES 4
of Z1(24 X) . The quotient space R1(21, X) = 2' (Il, X)/W (ll, X) is called
the first cohomology g ~ o u p of 2i with coefficients in X. Al1 the amenability
theories are concerned with the question of whether %'(a, X ) = {O} for
specifically chosen classes of Banach %-modules.
A Banach algebra II is said to be contractible if %'(a, X ) = {O} for
al1 Banach Il-b-birnodules X, amenable if U1(U, X*) = {O) for al1 Banach
U-bimodules X, and weakly aamenable if 3C1 (Z, a*) = {O). A contractible
Banach algebra has an identity and an amenable Banach algebra has a b.a.i.
((161 and [49]).
A contractible Banach algebra is finite dimensional if it has the bounded
approximation property [67] or if it is commutative [16]. We do not know
that a contractible Banach algebra is always finite dimensional or not. We
refer to [39, VI1 5 1.4-1-51 as well as [67] and [65] for properties of con-
tractibility.
Amenability theory of Banach algebras was originated by B. E. Johnson
in [49], where he proved that a locally compact group G is amenable in the
classical sense (see [19] and [27]) if and only if the Banach algebra L1(G)
with the convolution product is amenable. Other approaches to this concept
have been discussed in (471 and [39, Chapter VII]. For second dual algebras
of group algebras we know that L1(G)** is amenable if and only if G is a
finite group [26]. Another remarkable result is that a C*-algebra is amenable
1.1. PRELIMINAR,IES 5
if and only if it is nuclear, proved by U- Haagerup and A. Connes in [36]
and [12].
Suppose that M is a subset of a Banach space X. We denote by M L the
annihilator of M in X*, i.e.
M~ = {f E X*; f l M = 0)
We will use the following result in this thesis.
THEOREM 1.3 ([16],Theorem 3.7). Suppose that 'U is an arnenable Ba-
nach algebra. Let J # O be a closed left, right or two-sided ideal i n %. Then
J has a right, left, or two-sided b.a.i . if and only if JL i s complemented
in II' (Le., J is wealily complemented: see page 96 for the definition of weak
complementabilit y.).
It has been conjectured that every reflexive, amenable Banach algebra
is finite dimensional. In 1992 J. E. Gale, T. J. Ransford and M. C. White
proved that this is true if irreducible representations of U are finite dimen-
sional [25]. This resuIt was much improved in [44] by Johnson, who showed
that this is the case if each maximal left ideal of U is complemented. In 1996
F. Ghahramani, R. J. Loy and G. A. Willis proved that this is true if the
underlying space of is a Hilbert space [26]. Although this result is covered
by Johnson's preceding result, the method is totally different and is of special
interest. Recently, V. Runde [61] proved a new result: if each maximal ideal
1.1. PRELIMINARIES 6
of U is finite codimensional then the conjecture is true. In Chapter 6, we
will prove a theorem which improves both Johnson's and Runde's results.
W. G. Bade, P. C. Curtis and H. G. Dales first introduced the concept
of weak amenability in [2] for commutative Banach algebras. The origi-
nal definition used there is that a commutative Banach algebra 2.t is weak?y
amenable if ?il(%, X) = {O) for aLl symmetric Banach Il-bimodules X. They
proved that this is actually equivalent to R1(ll, Il*) = (O). Using this equiva-
lence, Johnson called a Banach algebra (not necessarily commutative) weakly
amenable if 3t1(%, a') = {O) [46].
In the rest of this section, we will mention several Banach algebras of
concrete types. Their definitions can be found in the mentioned references.
Since t hey are not the objects of this thesis, we will not give formal definitions
for thern.
It is known that the list of weakly amenable Banach algebras includes
L1(G) [45] [20], Cc-algebras [36], the tensor algebra E&E* of any Banach
space E [l?], and some Lipschitz algebras [2]. By using a result of [9]
Ghahramani, Loy and Wülis proved in [26] that both M ( G ) and L1(G)** fail
to be weakly amenable for al1 infinite, nondiscrete abelian locally compact
groups G. Other related results can be found in [52]. Many interesting
properties of weak arnenability have also been investigated by N. Gronbæk
in (301-[33].
1.1. PRELIMINARIES 7
In [17], Dales, Ghahramani and Grgnbæk introduced the concept of n-
weak amenability as stated below.
DEFINITION 1.4. A Banach algebra 2l is n-weakly amenable, n 2 O, if
( ) = {O), and is permanently weakly amenable if it is n-weakly
amenable for al1 n > 1.
It is pointed out in [17] that (n + 2)-weak amenability implies n-weak
amenability for n 2 1, and if U is an ideal in %** then weak amenability of U
implies (272 + 1)-weak amenability. A variety of concrete algebras have been
investigated there: L1(G) is (2n + 1)-weakly amenable for al1 n (later John-
son proved that ZL(F2) is permanently weakly amenable [48]); C*- algebras
are permanently weakly amenable; for each infinite compact metric space
K, lipaK is permanently weakly amenable if a < 112, and is 2m-weakly
amenable if O < a < 1, but lip,ïï is not weakly amenable for a > 112;
N(E), the algebra of nuclear operators on E with E a reflexive Banach
space having the approximation property, is (2n + 1)-weakly amenable but
not Pn-weakly amenable. An open question raised there is as follows.
QUESTION 1.5. Does weak amenability imply bweak amenability?
After an extensive study of the Banach algebra U X in Chapter 4, we
will answer this question in Chapter 5; the answer i s negative.
Since there exists a permanently weakly amenable Banach algebra of the
form 2l b X (see Remark 4-36), a permanently weakly amenable Banach
1.2. OUTLINE OF THE CONTENT 8
algebra may have a nilpotent ideal that is complemented. This is not the
case in amenable Banach algebras. In fact, from Theorem 1.3 one can see
that non-zero nilpotent ideals in amenable Banach algebras, if any, can not
be weakly complemented. It is known that there do exist amenable Ba-
nach algebras that contain non-zero nilpotent ideals [68]. Loy and Willis
in [54] proved that for biprojective Banach algebras with a central approxi-
mate identity any non-zero nilpotent ideal can not have the approximation
property. Here a Banach algebra II is biprojective if the continuous bimod-
ule morphism n: ~ 6 % + a, specified by *(a 8 b) = ab , has a continuous
bimodule morphism right inverse T: U , i e n 0 T = la, the iden-
tity operator on II, where %&.4 denotes the projective tensor product. One
can see [64] for further properties of biprojective Banach algebras. A con-
tractible Banach algebra is in fact a biprojective Banach algebra with an
identity ([39, VI1 51.41). We will introduce the concept of approximate com-
plementability for subspaces of Banach spaces in Chapter 7 and improve
Loy's and Willisl above result by showing that non-zero nilpotent ideals in a
biprojective Banach algebra with both left and right approximate identities
can not be approximately cornplemented.
1.2. Outline of the content
In Chapter 2, we construct, for any Banach algebra !2l and 2-bimodule
X, the %(2m)-bimodule actions on x (*~) for m > 1, assuming 11 (~~) being
the 2mth dual algebra of II equipped with the Awns first product. This
1.2. OUTLINE OF THE CONTENT 9
discussion sets a cornerstone for the three Chapters following it, especially
Chapter 4 and 5. Chapter 3 concerns the general theory about n-weak
amenability of Banach algebras. We will improve or extend some crucial
results of [17]. In Chapter 4 we discuss the module extensions of Banach
algebras, give a necessary and sufficient condition for them to be n-weakly
amenable, and consider several important special cases. Concrete exam-
ples will also be discussed there illustrating the relations between the weak
amenability of different orders. In the chapter following it we are construct-
ing an example of a weakly amenable Banach algebra which is not 3-weakly
amenable, answering the open question mentioned in the preceding section.
Chapter 6 deals with reflexive amenable Banach algebras. We will prove a
theorem which improves Johnson's theorem in [44] and Runde's theorem in
[61]. In the last Chapter, Chapter 7, we consider nilpotent ideals. We will
give the concept of approximate complementability and discuss the relations
between it and other well-known concepts such as weak complementability
and the approximation property. Then we wilI consider approximately bipro-
jective Banach algebras, showing that this kind of algebra can not have a
non-trivial nilpotent ideal that is approximately complemented.
CHAPTER 2
Dual Modules
2.1. Arens products
Suppose that % is a Banach algebra. In [l] Arens defùied two Banach
algebra products on II", the second conjugate space of 2, each extending
the product of II as canonically embedded in a**. The first (or left) Arens
product (a, Q) i+ W?: U** x %** + U** is given by
f = f ) (f E II*)
where @ f E IU' is defined by
The second (or right) Arens product @ - !P on %** is given by
where f - E U* is defined by
These products can be different for some Z. If <PP = B . %P for all
@, XV E II**, then 24 is called Arens regular. L1 (G) is Arens regular only
10
2.2. BIMODULE ACTIONS OF ON x ( ~ ~ ) I l
when G is a finite group [70], while a C*-algebra is always Arens regular
[Il]. See the survey article [22] for hirther properties of Arens products.
Viewing u ( ~ ~ ~ ~ ) a s the second dual of ~ ( ~ " 1 , m 2 O, we can define Arens
products in for each n 2 1 inductively. In this thesis we always take
the first Arens product for al1 the 2nth dual algebras Z(2n). If p, v E
such that p = wk*- lima,, u = wkr- limup for nets (u,), ( v ~ ) C ~ ( ~ " 1 , then
we have the following:
pu = wk*- lim lim u,u~. Q B
2.2. Bimodule actions of on x ( ~ )
Suppose that II is a Banach algebra, and X is a Banach Il-bimodule.
Then according to [17, pp. 27-28], X*' is a Banach Il**-bimodule. The
module actions are successively defined as follows:
First, for x E X, f E X*, 4 E X** and u E W* define $f; fx E 21* and
uf E X* by
(x E X).
Shen for 4 E X*', u E U*', define u4, $IL E X** by
They give the bimodule actions of a** on X**. Nso the definition for u f
with u E %** and f E X* gives a left Banach module action of a** on X*.
2.2. BIMODULE ACTIONS OF !Zt2") ON .X(lrn) 12
When u = a E 8, these module actions agree with the module actions of 2l
on the corresponding dual modules X* and X".
Viewing as a new II and X(2m) as a new X, the preceding procedure
will successively define X(2m+2) to be a Banach ~ ( * ~ ~ * ) - b i m o d u l e for m > 0.
So ~ ( ~ " 1 is naturally a Banach ~ ( ~ ~ ) - b i m o d u l e for al1 non-negative integers
m. Since some relations arising from this procedure are important for later
use, we now give the definition in detail as follows.
Suppose that the bimodule actions of on x ( ~ ~ ) have been de-
fined, where rn 3 O. Then in a natural way, x ( ~ ~ + ~ ) , k 2 l, i~ a Banach
~ ( ~ ~ ) - b i m o d u l e with the following module actions:
For A E X(2mf k, and u E z ( ~ ~ ) , U A , Au E x ( ~ ~ ~ k, are defined by
If u = a E a: these module actions agree with ?L-bimodule actions on
~ ( 2 m f k )
Then define, for F E x(*"+') and ip E x ( ~ ~ + ~ ) , F q , @F E u ( ~ ~ + ' ) by
(u , F a ) = (F , Gu) (= (uF, @))
and
(u, Q F ) = (FU, @) (= ( F , uQ)), (u E u ( ~ ~ ) ) .
2.2. BIMODULE ACTIONS OF 2fc2") ON 13
Throughout this thesis, for any Banach space Y and an element y E Y, ij
always denotes the canonical image of y in Y** (but to avoid unnecessary
complicated notations, we often use the same notation y to represent its
canonical image in any 2mth dual space When F E x ( ~ ~ + ' ) and
# E x ( ~ ~ ) , we denote F$ by Fq$ and $8' by 43'. It is easy to check that
(2-1) (u7 Fq5) = (&, F ) , (u; 4 F ) = (u4, F ) f o r u € ~ ~ ( ' ~ ) .
By using the canonical image of F or 9 in the appropriate 21th dual space,
we can then signi@ a meaning to F<P and <PF for any F E x ( ~ ~ + ' ) and
@ E ~ ( ~ " 1 . They are elements of Z(~"C, where k = max{m, n - 1).
Now for p E 5X(2mf 2, and F E x(*~+'), define pF E x(~"+') by
This actually defines a left Banach ~ ( ~ " ~ ~ j - r n o d u l e action on x (~~+ ' ) .
Finally, for /I E and E x ( ~ ~ ~ ~ ) , define p9 , @ p E by
( F ) = ( F , p), ( F ) = ( F ) (F E x ( ~ " + ~ ) ) .
They give the ZL(2m+2)-bimodule actions on x ( ~ ~ + ~ ) and complete our defi-
nit ion.
If lim u, = p in ~ ( I u ( ~ ~ + ~ ) , !$2m+1) ) and lim 4o = in a (~(2m+2) , x ( ~ ~ + ' ) ) ,
where (u,) c z(~") and (#f l ) c ~ ( ~ ~ 1 , then
2.2. BIMODULE ACTIONS OF a('") ON x(~")
For p E 21(2mC2) and 4 E ~ ( ~ " 1 , since pQ = p6, 4p = &L we have
One can also easily check the following relations:
where f E x ( ~ ~ - ~ ) , # E x ( ~ ~ - ~ ) , Q E x ( ~ ~ ) and u E (m 2 1).
Therefore each product agrees with those previously defined.
Concerning dual module rnorphisms we have the following.
PROPOSITION 2.1. Suppose that X and Y are Banach U-bimodules. Then
for each integer m 2 1 and each continuous U-bimodule morphism T : X + Y ,
r(2m) : x ( ~ ~ ) + Y(2m), the 2mth dual operator of T , is an 2L(2m)-bimodde
morphisrn.
PROOF. It suffices to prove the proposition in the case rn = 1. Suppose
that @ E X**, u E U**, and
2.3. BIMODULE ACTIONS OF !IlZL('*) ON x ( ~ ~ )
where (a,) c II, (xg) c X. Since T** is weak* continuous, we have
T" (@a) = T** (wk*- lirn lim 5Pâ,) = wk*- lirn lirn T** (Zpâa) P Q P Q
= wk*- lim lim T ( x ~ , ) ^ = wk*- lim lim T ( x P ) ^ ~ ~ B Q B a
= wk*- lirn lim T** (ZB)âa = T** (@)u. B Q
Similar calculations show the identity for the Ieft action. Cl
Concerning derivations we have the following proposition which is an
extension of [17, Proposition 1.71.
PROPOSITION 2.2. Suppose that 2t is a Banach algebra and X is a Ba-
nach %-bimodule. If D: % + X is a continuous derivation, then for n > O
D(*"): 21(2n) + x(*~), the 2nth dual operator of D , is a h a continuous
derivation.
PROOF. For n = 1, the proof is similar to the proof of the preceding
proposition, and by induction on n the proof can be easily cornpleted. O
CHAPTER 3
The n-Weak Amenability of Banach Algebras
The theory of n-weak amenability of Banach algebras was established
by H. G. Dales, F. Ghahramani and N. Gr~nbæk in [17]. It extends the
theory of weak amenability. In Chapter 1 we already gave the definition and
narrated the backgroud of this concept.
3.1. (n + 2)-Weak amenability and n-weak amenability
In this thesis We use the notation + for the direct sum of Banach spaces,
or the direct sum of modules, while we Save the notation @ for module
extentions of Banach algebras.
Suppose that D: U + U(") is a continuous derivation, n 3 1. Since
is an U-bimodule decomposition, D can be viewed as a continuous derivation
from fU into a("+ 2 ) . If SL is (n i 2)-weakly anienable then D is inner in SL("+*)-
By considering the U-bimodule projection from u("+~) onto a(") it foliows
that D is imer in a("), showing that P is n-weakly amenable. This was
observed by Dales, Ghahramani and Granbaek. So we have:
3.1. (n + 2)-WEAK AND n-WEAK 17
PROPOSITION 3.1 ([l?], Proposition 1.2). Suppose that 24 is a n (n + 2)-weakly
amenable Banach algebra, n >_ 1. Then U is n-weakly amenable.
From this proposition any (2m + 1)-weakly arnenable Banach algebra
must be weakly amenable. It is of course interesting to know if the converse
is also true. This question was raised and left open in [17]. We will give
a negative answer to it in Chapter 5. Here we are interested in conditions
that can guarantee the converse. Before this discussion we give an important
property of weak amenability, which is taken from [17] (see also [32] for the
commutative algebra version). We include the proof for completeness.
PROPOSITION 3.2 ([17], Proposition 1.3). Svppose that % is a w e a h amenable
Banach algebra. Then 11* = span{ab; a, b E Q) is dense in II.
PROOF. Let X E 24' satisfy Xlaz = O. Let D: Q -t II* be the bounded
operator given by
Then D is a derivation and hence is imer. There is an f E 8' such that
Taking b = a, we have (a, A)* = O for al1 a E Q. So X must be O, which
implies that IU2 is dense in 2i by the Hahn-Banach Theorem. O
3.1. (n + 2)-W3AK AND n-WEAK 18
LEMMA 3.3. Suppose that 2l 2s a lefi (right) ideal in a". Then it 2s also
a lef t (resp. right) ideal in for al1 rn 2 1.
PROOF. We show that if !Z is a left (right) ideal of rn 2 1, then it
is a left (resp. right) ideal of 21(2m+2). Then by using induction we will have
the conclusion.
In fact ,
and
as I21-bimodules. For any F E 21(2mf '1, write F = fi i A, where fi E and
f2 E T. If 2l is a left (right) ideal of M(~""), then a fi = O (resp. fia = 0) for
a E U. So
aF =af2 = @fi)^ (resp. Fa = f2a = (fi a)*) .
For any E %(2m+2), let = 4 + fi, where # E (W)I and u E W* . Then
(F , @a) = ((ah); 6) = (F, ( ~ 4 ~ )
(resp. (F, a@) = (F, (au)^)).
So <Pa = @a)'% a (resp. a@ = (au)^€ a) for a E %. Thus is a left (resp.
right) ideal of 21(2m+2). The proof is complete. O
3.1. (n + 2)-WEAK AND n-WEAK 19
Remark 3.4. Suppose that b is a Banach algebra and U = W*. If
23 is an ideal in B**, then the natural way to make b an IU-bimodule is
using the Arens product to give the module actions; but, when we go to
higher duals, the distinction between algebra products and module actions
makes superficial sense. For instance, since 23 is an ?L-birnodule, B** is an
a"-bimodule; for b E B c 113" and î~ E %**, the module actions u - b, b - u
make sense and give elements of B". Since 23 C B** c B(*), ub, bu also
make sense and are elements in B(*) thanks to the Arens product in ~ ( ~ 1 .
But from the above lemma, ub,bu E 23 c b**. It is routine to check that
as elements in B**, u b = ub and b . u = bu. This fact will be used later in
Chapter 5.
THEOREM 3.5 ([17], Proposition 1.13). Suppose that 24 is a weakly amenable
Banach algebra and zs an ideal in %**. Then 'U is (2m + 1)-weakly amenable
for al1 m 2 1.
PROOF. Since (u, a F ) = (ua, F ) and (u, Fa) = (au, F ) for u E %Z2"),
a E 2l and F E %(2mf1), from Lemma 3.3, aF = Fa = O for al1 a E U and
F E nL n Suppose that D: a+ is a continuous derivation
and P: + UL is the U-module projection with the kernel 8' (see
equation (3.2)). Then
3.1. (n -t 2)-WE.U AND n-WEM 20
This shows that P O D = O from Proposition 3.2. Therefore D is in fact a
derivation from 5X into U*. So D is inner from the assurnption. It follows
that fU is (2772 + 1)-weakly amenable.
O
LEMMA 3.6. Suppose that 2i is a Banach algebra vith a left (right) b.a.i. .
Suppose that X is a Banach 8-bimodule and Y is a weak* closed submodzrle
O/ the dval module X*. If the lep (resp. right) 2i-module action on Y is
trivial, then R1(ll, Y ) = {O).
PROOF. We prove the result only in the case 'U has a left b.a.i.. The
proof for the other case is similar. Suppose that D: II + Y is a continuous
derivation. Let (ei) be a left b.a.i. of II, and f E Y be a weak* cluster point
of (D(ei)). Since %Y = {O), we have
D(a )= l imD(e ia )= f a = f a - a f , ~ € 2 4 -
Hence D is inner, showing that ?il(%, Y) = {O).
When Y = X* and the right (left) U-module action on X is trivial, the
above lemma gives the following result due to Johnson (we indicate that
in Johnson's origional result the condition of the exsistence of a b.a.i. is
unnecessary and can be replaced by an appropriate one-sided b.a.i.).
3.2. T m n-WBAK AMENABLITY OF 21d 21
COROLLARY 3.7 ([49], Proposition 1.5). Suppose that P is a Banach al-
gebm with a left (right) b.a.i. and X is a Banach Il-bzmodule with the right
(resp. lejt) module action trivial. Then 'ftL (a, X*) = {O).
With Lemma 3.6, we can pruve another partial converse result to Propo-
sition 3.1 as follows.
THEOREM 3.8. Suppose that U is a weakly amenable Banach algebra. I j
II has a left (right) b.a.i. and is a left (resp. right) ideal in U", then 2 is
(2m + 1)-weakly amenable for rn 2 1.
PROOF. We prove the result only in the case that II has a left b.a.i. and
is a left ideal in II". From the It-bimodule decomposition (3.2) we have the
cohomoIogy group decomposition
If 2 is weakly amenable, we have 711(<U, II*) = {O). 21L is clearly weak*
closed submodule of Since IZI is a left ideal in 8", it is a left ideal
in u ( ~ ~ ) from Lemma 3.3. It follows that the left U-module action on UL is
trivial. Then Lernma 3.6 leads to that ?L'(a, a'-) = {O). As a consequence
we have X1(U, 11(2mCi)) = {O) and su U is (2772 + 1)-weakly amenable. O
3.2. The n-weak amenability of IId
Suppose that II is a Banach algebra. Let '@ be the unitization of U, the
Banach algebra formed by adjoining an identity e to fU, so that UV = %+Ce
3.2. THE n-WEAK AMENABZLITY OF 2tC
with the product given by
Here we use + instead of @ to denote the direct sum, because the latter
will have special meaning in this thesis. In this section we are interested
in finding the relations between the weak amenability of I( and II" If U
is commutative, then 2 L g is n-weakly amenable if and only if II is n-weakly
amenable ([17]). In the general case things seem complicated. It is known
that if 2l is (Zn + 1)-weakly amenable then so is Q< and if 2@ is Zn-weakly
amenable then so is 2l ([17]). The converses remain open [28].
It is easy to see that
where n 2 0, and e* E %O* satisfies e*lm = O and (e, e*) = 1. The first Arens
product on is given by
and the module operations of 2Lfl(2n) on ~ g ( ~ ~ + ~ ) for m 2 n 2 O are given by
(U + a e ) ( F + Be*) = (uF + O F ) + ((u, F ) + a&*,
( F + Be*) (u + a e ) = ( F u + aF) + ( ( 7 4 F ) + @)e*,
3.2. THE n-WAK AMENABILITY OF 21n 23
for u E F E z (~~+' ) , and a, P E C. Especially the nu-bimodule actions
on ~ f l ( ~ ~ + ' ) are given by the formulas:
(3-3) (a + a e ) ( F + Be') = (aF + aF) + ( (a , F ) + aO)e*,
(3 -4) ( F + Be*) (a + a e ) = (Fa + aF) + ( (a , F ) + @)e*,
for a E U, F E 21(2mf1)1 and crlP E @.
PROPOSITION 3.9. For rn 2 O , 2@ is (277-2 + 1)-weakly amenable if and
only i f 'LLZ 2s dense in II and every bounded derivation D: U t with
the condition that there zs a T E W such that
i s inner. If this is the case, then T = 0.
PROOF. For necessity, assume 24* is (2m + 1)-weakly amenable. Let
f E 8' be such that f 1 % ~ = 0, and let A: %fl + 21d(2mf '1 be defined by
A(a + a e ) = (a , f )e* (a E %,ci. E c).
Then A is a continuous derivation. So A is inner. A simple calculation by
using formulas (3.3) and (3.4) shows that f = O. Therefore Q12 is dense in
- Suppose that D: 2l + '1 is a bounded derivation satisfying (3.5).
Then D: Ufl + 1("2m+1) defined by
3.2. THE n-WEAK AMENABILITY OF 2Ln
is a continuous derivation. In fact ,
- D((a + ae) (b + Be) ) = D (ab + a b + Ba) + (ab + a b + Da, T)e*
= (D(a) + (a, T) e*) - (b + pe) + (a + cre) (D (b) + (b, T) e*)
- = D ( a + ae) ( b +De) + (a+ ae) -D(b+pe) .
So D is inner. It then follows that D is imer and T = 0.
For the sufficiency, let A: ?LI be a continuous derivation.
Since A(e) = O we can assume
where D: 2l + -t(2m+1) is a bounded operator and T E a'. We have from
formulas (3.3) and (3.4)
= aD(b) + D(a)b+ ( (a ,D(b)) + (b, D(a)))e* (a, b E a).
This shows that D is a derivation from U into IZL(~~+ ' ) and T satisfies (3.5).
Thus D is inner, which in turn implies that (ab, T) = O for al1 a, b E U. So
T = O. It follows that A is inner. So Uu is (2m + 1)-weakly amenable. The
proof is complete. O
COROLLARY 3.10. Suppose that II has a b.a.i.. Then for rn 2 O the
algebra IlD is (2m + 1)-weakly amenable if and only if M is (2m + 1)-wea&
amenable.
3.2. THE n - W E A K AMENABILITY OF 'iUU 25
PROOF. Let (ei) be a b.a.i. of ll, and let E E b e a weak* cluster
point of (ei). For any continuous derivation D: 2l+ N- ' m f '1, define T E a*
(a, T) = (D(a) , E ) , a E Il.
Therefore Equation (3.5) holds for any continuous derivation D: Il+ l).
The rest is clear.
Remark 3.11. For rn = O Corollary 3.10 was obtained by Gronbæk in
[3Ol.
COROLLARY 3.12. Suppose tha t 2t is a dweakly amenable Banach algebra.
Then is (2m + 1)-weakly amenable if and only if Il is (2m + 1)-weaklg
amenable.
PROOF. We only need to prove the necessity. Suppose tha t D: II-+
is a continuous derivation. Let P: Il(*"+') + U* be the projection with the
kernel a". Then P O D: U + U' is an inner derivation. On the other hand,
the bounded derivation (1 - P) 0 D: 2l -t II1 satisfies Equation (3.5) with
T = O. From Proposition 3.9, (1 - P) O D is inner. This shows that D is
inner. So II is (2m + 1)-weakly amenable. 0
3.2. THE n-WEAK AMENABILITY OF 21d 26
It was shown in [17, Proposition 1.41 that if QD is 2m-weakly amenable,
m 2 O, then U is Sm-weakly arnenable. For the converse we have the follow-
ing:
PROPOSITION 3.13. Suppose that U is 2m-weakly amenable, m > 0 , and
212 is dense in U. Then II$ is am-weakly arnenable.
PROOF. Let A: Uf' + 2lU2") be a continuous derivation. Then there is
f E !W and a bounded operator O: U -t 11(2m) such that
A ( a + cue) = D(a) + (a , f)e, a E a, a E C.
Routine calculations lead to the bllowing equalities:
D(ab) = aD(b) + D(a)b + (b, f ) a + (a, f )b7
(ab, f ) = O, a, b E U.
The 1 s t equality shows that f = O since U2 is dense in a. It follows that D
is a derivation and hence it is inner, and so there is 4 E U(2m) such that
Thus A is inner and consequently QI is 2m-weakly amenable. O
From Cohen factorization theorem (Theorem 1.1) and using the above
proposition, we have immediately the following.
COROLLARY 3.14. If has a b.a.i . , then Ut is 2m-weakly amenable if
and only if U is 2m-weakly arnenable.
3.2. THE n-WEAK AMENABILITY OF 2Ld 27
From Proposition 3.2, Proposition 3.13 and [l?, Proposition 1.41 we ob-
tain also the following:
COROLLARY 3.15. Suppose that II is weakly amenable. Then 2Lfl zs 2m-weakly
amenable i f and only i f % is Sm-weakly amenable.
To end this Chapter we combine corolaries 3.10, 3.12, 3.14 and 3.15 to
a theorem as follows.
THEOREM 3.16. Suppose that 2i is a Banach algebra tuhich is either
weakly amenable or has a b.a.i. . Then for each n 2 O , i s n-weakly
amenable if and only i f 2l is n-weakly amertable.
CHAPTER 4
The n-Weak Amenability of Module Extensions of
Banach Algebras
Suppose that II is a Banach algebra and X is a Banach a-bimodule. The
module extension algebra @ X is the Il-direct sum of 'LL and X, with the
algebra product defined as follows:
(a, X) . (b, y) = (ab, ay + xb), (a, b E II, x, y E X).
It is a Banach algebra. Some properties of this kind of algebra have been
discussed in [3] and [17]. In this chapter we study the n-weak amenability
of this kind of Banach algebra. Since X is a complemented nilpotent ideal of
U $ X, according to Sheorem 1.3, U @ X can never be an amenable Banach
algebra unless X = O. If 24@X has both left and right approximate identities
(for instance, when 2l has both left and right approximate identities and
they are also, respectively, left and right approximate identities for X) , then
24 @ X can not be pointwise approximately biprojective (see Definition 7.10
and Theorem 7.14). But U Q X can be weakly amenable. It may even
be permanently weakly amenable. In section 4.4 we will see that if 24 is a
weakly amenable commutative Banach algebra with a b.a.i., then fil@ A. is
28
4. MODULE EXTENSIONS OF BANACH .LUGEBRAS 29
perrnanently weakly amenable, where A. is identical to 24 as a left ILI-module
while the right module action on A. is trivial.
This chapter is organized as follows: In Section 4.1 we first discuss the
(P @ X)-module (TU @ x)("), giving the formulas for the module actions.
Then we d l discuss various techniques for lifting derïvations. In Section 4.2
we prove two theorems which give the necessary and suficient conditions
for (3 @ X) to be n-weakly amenable. Sections 4.3 and 4.4 deal with the
special cases of X = U and X = Xo, where Xo denotes a Banach U-bimodule
with the right module action trivial. We will encounter severai interesting
examples there. This chapter is also a foundation for Chapter 5, where
we construct a counter-example to answer question 1.5 in Chapter 1 in the
negative.
Sometimes we regard IU @ X as an Il-bimodule with the natural module
actions: b - (a, x) = (ba, bx) and ( a , x) . b = (ab, xb) for b E a, ( a , x) E X.
Since the dual module (M @ X)* is identical with ( o + X ) ~ + ( U + O ) ~ , where
the sum is 2,-sum, and (0+X)'- (resp. (%+O)-'-) is isometrically isomorphic
to IU* (resp. X*) as SI-bimodules, for convenience in this thesis we just write
(ne X)' = W+X*.
Similarly we will identify the n th dual (I( @ x)(") with %(") /x(*), where
the sum is 1,-sum when n is odd, and is Il-sum when n is even. For the
4-1. LIFTMG DERIVATIONS 30
moment this identity is in the sense of rU-bimodules. We will study the
(II @ X)-bimodule actions in the next section.
4.1. Lifting derivat ions
In this section we give several lemmas concerning lifting derivations (resp.
module morphisms) from II (resp. X) into the modules u(") or x(") to
derivations from @ X into ('U @ x)(~). We first discuss the (24@ X)-module
actions on the nth dual module (II @ x)("). Recall that, when n is an odd
number, the notations XX(") and x(")x have been defined in Chapter 2 on
page 12 for x E X.
LEMMA 4.1. Szppose that X is a Banach %-bimodule. Then the (0 @ X ) -bimodule
actions on (II @ x)(") are given by the following jormulas
PROOF. We use induction to give the proof. For n = O the equalities are
just the definition of the algebra multiplications for 2l@ X. Assume that
4.1. LIFTING DERIVATIONS 3 1
they are true for n = 2m1 m 2 O. Then for (a(2m), x ( ~ " ) ) E (Z û3 X)(2B) ,
(a(2m+l) 2(2m+l)) E (a ~ ) ( 2 m + l )
and
These show that the equalities hold for n = 2.m f 1.
4.1. LIFTING DERIV.4TIONS 32
Now assume that they are true for n = 2m - 1, rn 2 1. Then for
(a(2m-l) , x(2m-l) ) E (a @ x)(~~- ' ) and (a(2m), x ( * ~ ) ) E (a @ x ) ( ~ ~ ) ,
and
So the equalities hold also for n = 2m.
By induction, we see that the equalities (4.1) and (4.2) hold for any
integer n 2 O. 0
We have known that for each integer rn, x ( ~ ~ ) is a Banach 11(2m)-bimodule.
So the module extension Banach aigebra 11(2m)@~(2m) makes sense. Also the
2mth dual algebra has the same underlying space as @ x ( ~ ~ ) .
4.1. LIFTING DERIVATIONS 33
The distinction between these two algebras is in fact superficial as stated in
the following remark.
Remark 4.2. The algebra @ x(*~) is isometrically isomorphic to
(a EI x) (2m) .
PROOF. 30th have the underlying space ~ ( ' ~ 1 +x(*~) . Using induction,
One sees easily that the algebra products defined on them are also the same.
O
Now we can give Our lifting results.
LEMMA 4.3. Suppose that m 2 O and Dr 2 l - t N(*~") i s a (continuous)
derivation. Then D: U @ X + (24 03 x ) ( ~ ~ + ' ) defined by
i s also a (continuous) derivation. D is inner i f and only i;fD is inner.
PROOF. It is routine to check that D is a (continuous) derivation if D is
SO.
If D is inner, then there is u E 11(2mfl) such that D(a) = au - ua. This
leads to
- D((a7 4) = (a, 4 . - (u, O) - (a, 4,
and so is inner.
4.1. LIFTING DERIVATIONS
If D is inner, then there is (u, F) E @ x ( ~ ~ + ' ) , such that
and so D ( a ) = a u - ua, for al1 a E P. This shows that D is inner. O
LEMMA 4.4. Suppose that D: 2.i + x ( * ~ ) is a (continuous) derivation.
Then D: II @ X + ( I I @ x)('~) defined by
is also a (continuous) derivation. D is inner if and only i f D is.
PROOF. The proof is similar to that of Lemma 4.3. O
LEMMA 4.5. Suppose thaf n 2 O and D: + x(") is a (continuous)
derivation. T h e n D*: x(*+') + a*, the dual operator of D , satisfies
for a E U and F E x(~").
PROOF. For any b E U,
(b , D * ( a F ) ) = (D(b)a , F ) = (D(ba) - b D ( a ) , F )
= (b , a D * ( F ) - D(a)F) ,
4.1. LIFTING DEW.4TIONS 35
and so D*(aF) = aD*(F) - (D(a) F) 1%. A similar argument will give the
second equality. O
LEMMA 4.6. Suppose that k > O is an znteger and D: 2i + X(") is a
- (continuozls) derivation. Then for m > 0 , D(~"+'): x ( ~ + ~ ~ ~ ' ) + '21(2m+L),
the (2m + 1)th dual operator of D, satisfies
for a E Z l and F E
PROOF. First, for m = O, Lemma 4.5 gives the result. For m > O, from
[17, Proposition 1.71, ~ ( ~ ~ 1 : -t X(ef2m) is a (continuous) derivation.
Then from Lemma 4.5,
s at isfies
and
for u E 11(2m), F E In particular when u = a E a, these give the
formulas of the lemma. a
4.2. LIFTING DERIVATIONS 36
LEMMA 4.7. For any integer 2m + 1 > O , suppose that D: II + x ( ~ ~ + ' )
is a (continuous) derivation. Then Dr 2i @ X + (a @ x ) ( ~ ~ + ' ) defined by
is a (continuow) derivation. Moreover,
1. ifD is inner, then so is D;
2. if D is inner, then a (continuous) derivation 5: II @ X +- (24 x ) ( ~ ~ + ' )
can be set so that D( (a , O ) ) = O for a E a, and D - D is inner.
PROOF. For a, b E Q and x , y E X , from Lemma 4.6 we have
- D((a ,x) - (b, y ) ) = B ( ( a b , ay + xb)) = ( - ~ ( ~ ~ + ' ) ( a y + xb), ab))
= ( - [ , ~ P m + l ) ( Y ) - (D(a)~) la(2- ) f D (2m") (z) b - ( x D (b) ) Imc lm, ] , D ( a ) b + a D (b ) )
- - (-[a ~ ( ~ ~ ~ ' 1 ( y ) - D (a ) y + D ( ~ ~ + ' ) ( x ) b - z D ( b ) ] , D ( a ) b + a D (b) )
= (-,@m+') ( y ) + xD(b) , aD(b)) + ( - D ( ~ ~ + ' ) ( X ) ~ + D(a)y , ~ ( a ) b )
= (a, 2) - (- D(~"+') ( y ) 7 D ( b ) ) + (- D ( ~ ~ ~ '1 (z) , ~ ( a ) ) (b , .y)
= (a, x) - m, Y ) ) + m a , 4) . (b, Y )
Therefore D is a (continuous) derivation.
If D is inner, then for some u E Q Pm+l) F E ~ ( 2 m + l ) ,
- (O, D ( a ) ) = D((a, O)) = (a, 0) (21, F ) - (u, F ) - (a, 0)
= (au - ua, a F - Fa) .
So D(a) = aF - Fa for all a E Il. Thus D is inner.
Conversely, if D is inner, then t here is F E x('"+ '1 such that D (a) = aF - Fa
for a E Ill. Let T: X -t 1((2m+1) be defined by
T(x) = - D ( * ~ + (x) - (xF - FI) (x E X)
and T: M @ X + (II @ X ) ( ~ ~ + I ) be defined by
( D - T ) ( (a , 5)) = (2F - F z , aF - F a )
for (a , x) E U X. Therefore D - T is an inner derivation. This in turn
irnplies that T is a (continuous) derivation. So D = T will satisS. al1 the
requirements. The proof is complete. Cl
LEMMA 4.8. Suppose that r: X -t z(*~+') is a continuous Il-bimodule
morphism. Then D: 2t CEI X + (3 x ) ( ~ ~ + ' ) defined b y
4.1. LIFTING DERINATIONS 38
is a continuous derivation. D is inner i f and only i f there is F E X(2m+L)
such that aF - F a = O for a E I I , and î(x) = XF - Fx.
PROOF. It is easy to check that D is a continuous derivation. If D is
inner, then there is (u, F) E 8 x(*~+'), such that
(r(x): 0) = D((O1x)) = (O, X) - (a, F ) - (a, F ) ( 0 , ~ )
= (xF - Fx, O).
This shows that r ( x ) = XF - Fx for x E X. Also
(0, 0) = D((a, O) ) = (a1 O) - (u, F ) - ( 7 4 F ) - (a1 O)
= (au - ua, aF - Fa).
We have aF - Fa = O for a E U.
Conversely, if there is F E such that aF - Fa = O for a E II,
and r(x) = I F - Fx, then
D((a, x)) = (r(x), 0) = (xF - Fx, a F - Fa)
= (a, 4 (O1 F ) - (O. F ) . (a, 4,
showing that D is inner.
LEMMA 4.9. Suppose that T : X + ~ ( ~ ~ ~ ' 1 i s a continuous fll-bimodule
morphisrn, satisfying x T ( y ) + T ( x ) y = O for al[ x, y E X . Then
4.1. LIFTING DEW4TIONS
defined by
is a continuous derivation. D is inner i f and only i f T = 0.
PROOF. The first statement is easy to check. If D is inner, then for some
F ) E ~ ( ? m + l ) @ ~ ( 2 m + l )
( O , T ( x ) ) = D((0, x)) = (0, X) (u, F ) - (u7 F ) ( 0 , ~ )
= ( x F - Fr, O ) , x E X.
Therefore T = O. The converse is trivial. O
LEMMA 4.10. Suppose that T : X -t I U ( ~ ~ ) is a (continuous) Il-bimodule
rnorphism, satisfying xT(y) + T(x)y = O in x ( ~ ~ ) for x , y E X . Then
- T : II @ X -t (a@ x)(~") defined by
is a (continuous) derivation. T is inner i f and only if T = 0.
PROOF. The proof is similar to that of Lemma 4.9. Cl
LEMMA 4.11. Suppose that T : X + x ( * ~ ) is a (continuous) li-bimodule
rnorphism. Then T: 2l@ X -t (a @ x ) ( ~ ~ ) defined by
4.2. ON n-WEM AMENABILITY OF 2i @ X 40
is a continuous derivation. T is inner if and only zf there zs some u E z( *~)
such that ua = au for a E I I , and T ( x ) = xu - ux for al1 x E X.
PROOF. The first statement is obviously true. If S is inner, then for
some (u, 4) E z ( ~ ~ ) @ x ( * ~ ) ,
= (O, xu - ux)
and
(0,O) = T((a, O)) = (a, 0) (74 4) - (u, 4) ( a , 0)
= (au - ua, a4 - 4a)-
Therefore, T(x) = su - u s and au - ua = 0.
For the converse, suppose that, for some u E satisfying az~ - ua = O
for al1 a E a, T ( x ) = xu - ux for x E X, then
- T((a, z)) = (O, T(x)) = (au - ua, s u - ux)
= (a,x) - (u,O) -(a$) (a ,x) .
So T is inner.
4.2. O n n-weak amenability of U X
Suppose that is a Banach algebra and X is a Banach 2Lbimodule. In
this section we are concerned with the conditions for the n-weak amenability
4.2. ON n-WEAK AMENABILZR OF !2l û3 X 4 1
of the module extension Banach algebra TU @ X. The discussion naturally
splits into two cases of odd integers n and even integers n. For odd n we
have the following main result .
THEOREM 4.12. For m 2 O , II @ X is (2m + 1)-rueakly amenable i f and
only i f the following conditions hold:
1. B is (2m + 1) -weakly amenable;
2. N1(U, x(~*+')) = { O ) ;
3. For any continuous Il-bimodule morphism r: X + I Z L ( ~ ~ + ' ) there is
F E x(*~+') such that aF - Fa = O for a E .2l and r(z) = XF - F x
for x E X ;
4. For any continuous 2l-bimodule morphisrn Tc X -t x ( ~ ~ + ' ) , if
i n U(2mt1) for all x, y E X, then T = 0.
PROOF. Denote by Al the projection from (II @ x ) ( ~ ~ + ' ) onto Il(2mf ' 1 ;
the kemel of Al is x ( ~ ~ + ' ) - Let A2 = 1-Al, the projection from (a @ x ) (*~+ ')
ont0 ~ ( ~ ~ + l ) . Both Ai and A2 are obviously a-bimodule morphisms. Let
TI: % + II @ X be the inclusion rnapping (i.e. .r1(a) = (a, O ) ) . It is easy to
see that T, is an algebra homomorphism.
We first prove the sufficiency. Suppose that conditions 1-4 hold. Sup-
pose that D: 2l@ X + (a @ x ) ( ~ ~ + ' ) is a continuous derivation. Shen
4.2. ON n-WEAK AMENABPLITY OF !2l@ X
Dor1: % + (a@ X) (2m+L) is a continuous derivation. In fact,
Then Al O D O rl: U + 11(*~+') and A, 0 D 0 TI: 2L + x(*"+ ') are continuous
derivations. By conditions 1 and 2, they are inner. This implies that D 0 TI
is inner. From lemmas 4.3 and 4.7,
is a continuous derivation and there is a derivation D: II @ X -+ (a CEI x)(~~+ ' ) ,
satisfying D((a, O)) = O for a E SL, such that D o 7, - D is inner. Also
Let 5 = D - + We have that 8 is a continuous derivation from
X into (Ba x) (~~+ ' ) satisfying D((a, O)) = O for a E II . On the other
hand,
4.2. ON n-WEAK AMENABILITY OF SL û3 X
and
Denote by T . : X + 2l d3 X the inclusion mapping given by r2 (x) = (0, x) ,
(x E X). Then 5 o 72: X + (Q @ x) (~~+' ) is a continuous <Il-bimodule
morphism. From condition 3 there is F E x ( ~ ~ ~ '1 such that
A, o D o ~ ~ ( x ) =IF - F z , and aF - F a = O
for x E X and a E 2L. Since
we have
( ~ ~ o D o r ~ ( z ) ) ~ + x ( ~ ~ o D o . r ~ ( ~ ) ) = O
for x, y E X. Frorn condition 4, A2 0 5 0 72 = O. SO
4.2. ON n-WEAK AMENABILITY OF IZLe X 44
This shows that 5 is inner. Then D = 8 -f- (0 O TI - 5) is inner. This
proves that @ X is (2m + 1)-weakly amenable.
Necessity: Suppose that 2leX is (2m + 1)-weakly amenable, and suppose
that D: Q + I(('~+') is a continuous derivation. Shen, frorn Lemma 4.3,
D is inner. This shows that ?Y1(%, '1) = {O). Similarly Lemma 4.7
implies that X L ( Q , x ( ~ ~ + ' ) ) = { O ) . Therefore conditions 1 and 2 hold.
Since any continuous derivation D: % @ X + (a @ x)('~+') is inner,
Lemma 4.8 gives condition 3, and Lemma 4.9 shows that condition 4 holds.
The proof is complete. 0
For even integers n we have the following theorern.
THEOREM 4.13. For rn 3 O , 8 @ X is 2rn-vealily amenable if and only
i f the following conditions hold:
1. For any continuous derivation D: 2L + a('*), i f there is a continu-
ous operator T : X -+ x(?") such that T ( a x ) = D(a)x + a T ( x ) and
T ( x a ) = xD(a) + T ( x ) a , for a E SL and x E X, then D is inner;
2. R1(12L, x(*~) ) = {O);
3. Any continzlous Q-birnodule rnorphism r: X + u ( ~ ~ ) satisfying
in ~ ( ~ ~ 1 , for x , y E X , is trivial;
4.2. ON n-WEAK AMENABLLISY OF Zl CB X 45
4. For any continuous Il-bimodule rnorphisrn Tc X -t x ( ~ ~ ) , there exists
u E u(*"), such that au = ua for a E I I , and T ( x ) = xu - ux for
x E X.
PROOF. Denote by TI and 7 2 the natural inclusion mappings from, re-
spectively, % and X into 9.L @ X, and by Ai and A2 the natural projections
from (8 @ x ) ( ~ ~ ) ont0 2L(**) and x ( ~ ~ ) , respectively. These are a- bimodule
morphisms.
We first prove the sufficiency. Suppose that the conditions 1-4 in Theo-
rem 4.13 hold. Let D: (a@ X) -+ (a @ x ) ( ~ ~ ) be a continuous derivation.
Then clearly Al O D O TI: U -+ 1((*") and A2 0 D 0 T I : U -t x ( ~ ~ ) are
continuous derivat ions.
Claim 1: O D O T - : x i z U ~ ~ ) is trivial.
Let l? = Al O D O 75. To prove Claim 1, by condition 3 it suffices to show
that r is an U-bimodule morphism sa t i skng xI'(y) + r ( x ) y = O in ~ ( ' ~ 1 for
x, 3 X. In fact,
4.2. ON n-FVEAK AMENABILITY OF B CB X
Therefore xI'(y) + r ( x ) y = O. On the other hand
Similarly, l ' (sa) = I'(x)a and so I' is an 2Lbimodule morphism. This verifies
Claim 1.
Now let T = a 2 0 D o ~ 2 : x + x ( * ~ ) , and Dl = A ~ D O T ~ :
Claim 2: T ( a x ) = Dl(a)x + aT(x) and T ( x a ) = xDl(a) + T ( x ) a for
a E U , x E X -
In fact, from Claim 1
Similarly, (O, T ( x a ) ) = ( O , xDl(a) + T(x )a ) , for a E a, x E X . So Claim 2 is
verified.
Therefore by condition 1, Dl = Al O D o TI is inner. Suppose that
u E u ( ~ ~ ) satisfies Dl (a) = au - ua for a E U. Take Tl: X -t x ( ~ ~ ) specified
by T l ( x ) = x u - u x for x E X. Then T - Ti: X + x ( ~ ~ ) is a continuous
4.2. ON n-WAK AMENABZLITY OF !ü X
U-bimodule morphism. In fact, from Claim 2, for a E Z and x E X,
(T - Tl)(ax) = T(ax) - Tl(ax) = (Dl(a)x + aT(x)) - (axu - uax)
= (au - ua)x + aT(x) - (axu - uax)
= a(ux - XU) + aT(x) = a(T - c ) ( x ) .
Similarly, T - Tl is a right %-module morphism.
From condition 4; there is a v E %(2m) such that av = va for a E X, and
(T - T l ) ( x ) = xv - vx for x E X. From Lemma 4.11
is an inner derivation.
Since A2 O D O rl: O i x(**) is a continuous derivation, it is inner by
condition 2. From Lemma 4.4
is also inner. From al1 the discussion above combined, we have
is inner.
Hence as a sum of three inner derivations, D is inner. This shows that
under conditions 1-4 of Theorem 4.13, U @ X is 2m-weakly amenable. So
the sufficiency is true.
Now we prove the necessity. Suppose that %@X is 2m-weakly amenable.
Let D: 24 + u ( ~ ~ ) be a derivation with the property given in condition 1.
Shen D: U @ X -+ (a @ x ) ( ~ ~ ) defined by
is a derivation. Therefore, there are u E and 4 E X(2m) such that
= (au - ua,ad - c$a+ (xu -ux)).
This implies that D(a) = au -ua for a E IU. Hence D is inner. So condition 1
holds. Also lemmas 4.4, 4.10 and 4.11 show, respectively, that conditions 2-4
hold. The proof is complete. O
4.2. ON n-WEAK AMENABILITY OF I1@ X 49
Condition 4 in Theorem 4.13 seems to be a very tough condition, if
we note that the inclusion mapping from X into X(2m) is an 2bbimodule
morphism. So there are not many 2m-weakly amenable Banach algebras of
the form a@ X. In the rest of the section we make some comments on the
conditions in Theorem 4.12.
Remark 4.14. When m = O, condition 3 in Theorem 4.12 is equivalent
to
3'. There is no non-zero continuous a-birnodule morphism r: X i WU'.
PROOF. If 3O holds, then choose F = O E X*. We see that condition 3
holds for m = O. Conversely suppose that condition 3 holds for rn = O and
r: X + a' is a continuous 24-bimodule morphism. Then there is an F E X'
with aF - Fa = O, such that r(x) = XF - Fx for z E X. So
(a, r(x)) = (a, x F - Fx) = (x, Fa - a F ) = O, a E 53, z. E X.
Therefore, r(x) = O for x E X. This shows that r = 0.
For the general case, condition 3 in Theorem 4.12 is equivalent to:
3? For any continuous %bimodule morphism r: X + U(2m+1), the range
r (X) is contained in 21L and there is G E U and r(x) = XG - Gx for
a: € X.
PROOF. It suffices to prove that 3 implies 3m. Suppose that 3 holds, i-e.
for some F E x ( ~ ~ + ' ) sat isFng a F - Fa = 0, r ( x ) = xF - Fx (x E X).
4.2. ON n-WEAK AMENABXITY OF 2t G3 X 50
Then clearly r(x) E 21L. Let F = f + G, where f E X*, G E XL. Since
aG - Ga E XI, we have af - fa E r (X*) n XL, where L ( X * ) denotes the
image of X* in x (~~+ ' ) under the canonical mapping. This implies that
a f - fa = O. Then aG - Ga = O in x(*~+'). From the preceding remark
the operator ri: X -t SL* defined by r,(x) = xf - fi is trivial. Therefore,
Here we hzve used the fact that xf = (z f )^and fi = (f x)^ (see page 14). O
PROPOSITION 4.15. If condition 4 of Theorem 4.12 holds, then
PROOF. Assume conversely that cl(%X + XI1) # X. Take a non-zero
element F E X* fi (UX + X%lL, and let T: X + X* be defined by
T (x) = F (x) F.
Since FIaxcxm = O, it is easy to see that T is a non-zero continuous
a-bimodule morphism and UT(X) = T(X)U = {O). Also for x, y E X,
- O in a* since T(X) c ( n X ) I n (X2l)l. This shows that xT(Y) = T ( ~ Y -
condition 4 of Theorem 4.12 does not hold for m = O. So it does not hold
for any m >_ 0. O
Remark 4.16. For m = O, condition 4 in Theorem 4.12 is equivalent to
the following-
4.2. ON n-LVEAK AMENXBZLITY OF 2t @ X 5 1
4'. cl(2lX + XU) = X and there is no non-zero 2Lbimodule morphism
T: X + X* s a t i s b n g ( x ' T ( y ) ) + (y , T ( x ) ) = 0, for x, y E X.
PROOF. If condition 4 in Theorem 4.12 holds, the previous proposition
implies that cl(llX + X U ) = X . If an a-bimodule rnorphism T: X + X'
satisfies
then for any a E U,
This shows that xT(y) + T ( x ) y = O for x, y E X . Therefore T = O and so 4O
holds.
Conversely, if 4O holds, and T: X + X* is a continuous IU-bimodule
rnorphism satisking x T ( y ) + T(x)y = O in W, then for any x E Q X + X B ,
x = axl + x2b, and y E X , we have
Since cl(llX + Xll) = X, this implies that (x, T(y)) + (y, T(x)) = O for
al1 x, y E X. ~ e n c e T = O and so condition 4 of Thsorem 4.12 holds for
m=o. a
4.3. THE ALGEBRA !X Ci3 2i
4.3. The algebra !Z @ I1
In this and the following sections we will investigate several concrete
situations. This section is concerned rnainly with the two cases X = 2l and
X = Il* as Banach U-bimodules.
First we note that if B is not amenable, then there is a Banach Il-bimodule
X, such that U1(2I7X*) # { O ) . From Theorem 4.12, for this X : U @ X is
not weakly amenable. In fact, if X = II*, I1@ X is never weakly amenable
as stated in the following proposition.
PROPOSITION 4.17. For any Banach algebra <U, U@X* is never n-weakly
amenable fo r any n 3 0.
PROOF. From Proposition 3.1, it suffices to prove the proposition for the
cases of n = 1 and n = 2772, where rn 2 O. Notice that Condition 3' does not
hold, because the identity mapping from X (which is 2' in this case) onto
2P is a non-zero continuous 24-bimodule morphism. So from Remark 4.14
the statement is true for n = 1.
Suppose that n = 2m. Let T: X + x ( ~ ~ ) be a continuous Il-bimodule
morphism characterized in Condition 4 of Theorem 4.13 with X = a*, then
T ( f ) E UL for f E X. In fact, for a E a,
( a , T ( f ) ) = (a, f - u - u . f ) = (af -fa, u)
= (f, u a - a u ) = 0.
4.3. THE ALGEBFL4 2i @ B 53
But when viewed as a subspace of x ( ~ ~ ) , X does not annihilate II, where the
latter is viewed as a subspace of x (~~- ' ) . This shows that, as an Il-bimodule
morphism, the inclusion mapping from X into does not satisfy Con-
dition 4 of Theorem 4.13. Therefore %@ IC* is not Sm-weakly amenable. O
Now we consider the case X = fU. To avoid any confusion, from now on
when we regard X as an a-b-bimodde, we will use the notation A instead of
fU. Sirnilarly, we use the notation A(") to denote the Il-bimodule a("). in
this case, since condition 4 in Theorem 4.13 never holds for any integer m
(the canonical embedding is a non-zero morphism), we have that 2l @ A is
never Sm-weakly amenable for any m > O. If U is commutative, the same
reason yields a little bit more extended result. Recal that an 2-bimodule
X is symmetric if ax = xa, for a E U, x f X.
PROPOSITION 4.15. Suppose that 2l is a commutative Banach algebra.
Shen for any non-zero syrnmetric 2-birnodule X , M @ X is not 2m-weakly
amenable.
PROOF. If X is symmetric, then we will have xu = us for u E 21(2m) and
x E X. Since the inclusion mapping is a non-trivial U-bimodule morphisme
Condition 4 in Theorem 4.13 does not hold. CI
But irl A can be weakly amenable. Before giving an example for this
let us discuss some relations concerning corresponding elements of A(") and
TU("). Suppose that Q E A(n). We denote the same element in by 4.
4.3. THE ALGEBRA 2i 03 2î 54
LEMMA 4.19. Suppose that II is a Banach algeb~a and let m 2 O . Then
for 4, $ E and F E A(2mC'),
PROOF. It is easy to check that these equalities hold for rn = O. Assume
that they hold for m = k. Then for m = k + 1, letting 4, $ E A(?~+*) and
f E we have
This shows that $ f = 4 f. So
Hence (&b)" = $4, 4,1C, E A ( ~ ~ + ~ ) . Sirnilady $4 = (44)". These in turn
verify the following equalities.
4.3. TETE ALGEBRA 2i @ 11
where #, T) E A ( * ~ + ~ ) , F E il(2ki3)- SO #F = @? Also
Hence ($FI- = @. So we have proved q5F = (4~)" = 6F for the case
rn = k + 1. A similar proof d l $ve F 4 = ( ~ 4 ) " = &. This cornpletes the
proof. O
A special case of Lemma 4.19 is the following group of equalities which
we will use in the proof of the next theorem.
- zF = (ZF)" = ZF, Fx = (F5)" = Fz,
for a E 2l, x E A, E A ( ~ ~ ) and F E A ( ~ ~ + ' ) . From these equalities, we
also see that for X = A and m 2 O the conditions 3 and 4 in Theorem 4.12
hold if and only if there is no non-zero U-bimodule morphism T from A
into A ( ~ ~ + ~ ) . Also we note that in the case X = A conditions 1 and 2 of
Theorem 4.12 are the same.
THEOREM 4.20. Szppose that 2i is a Banach algebra.
1. if span{ab - ba; a , b E M) is not dense in O, then 2iB A is not weakly
amenable;
2. if span{ab-bu; a, b E II} zs dense in U, then %@A is weakly amenable,
provided U is weakly amenable and has a b.a.i..
4.3. TRE ALGEBRA M 56
PROOF. By condition 1 of Theorern 4.12 without loss of generality we
can assume 2l is weakly amenable for both cases. If span{ab - bu; a, b E a}
is not dense in a, then there is f E II* such that f # O and (ab - ba, f) = O
for a, b E II. So a f = fa for a E U. Then T: A + a' defined by
is an U-bimodule morphism. According to Proposition 3.2, 2l2, the linear
span of all product elements ab, a , b E U, is dense in a. So there are a , b E II
such that (ab, f ) # O . This implies that T # O. Therefore in this case ZL@ A
is not weakly amenable due to
If span{ab - ba; a, b E U} is dense in II, and II has a b.a.i., Say (ei),
then for any given continuous U-bimoduie morphisrn T: A + U*, we have
T ( a ) = a f = fa , where f is a weak* cluster point of (T (e i ) ) . Therefore
f (ab - bu) = O for al1 a , b E Zi. This shows that f = O and hence T = O. So
conditions 3 and 4 in Theorem 4.12 hold for rn = O. The other two conditions
hold automatically for m = O. So by using Theorem 4.12 we see that the
second statement of the theorem is true. El
From the first statement of Theorem 4.20 we have the following corollary
immediately :
COROLLARY 4.21. For any commutative Banach algebra Zi, @ A is
never weakly amenable.
4.3. THE ALGEBRA 3L €9 5 7
Let 3t be an infinite dimensional Hilbert space. According to a classical
result due to Halmos, every element in B('?f) can be written as a sum of two
commutators (Theorem 8 of [37]; for furtheï results about commutators in
C*-algebras we refer to [5]-[8], [38] and [59]). Together with the fact that
B ( X ) has an identity and as a C*-algebra is weakly amenable ([36]), from
Theorem 4.20 we see that B(U) @ B ( X ) is weakly amenable. Later in this
section we will see that it is actually (2mf 1)-weakly amenabie for al1 m 2 0.
PROPOSITION 4.22. In condition 2 of Theorem 4.20 i f we fvrther assume
that
V = span{au - ua; ZL E W*, a E Z)
is not dense in ZîM** +%**a (ifII has an identity this is equivalent to saying
that V is not dense in a**), then A is not 3-zueakly amenable.
PROOF. In fact, in this case %**a cl (V) , since otherwise it would follow
that both W* and %**a are in cl(V) and then cl(V) > %a** +W*II, which
is a contradiction to the assumption that V is not dense in W** + U**U.
Hence from the Hahn-Banach Theorem, there is F f W**, such that
FIV = O but F # O on %**Z. This implies that aF = Fa for al1 a E 2 and
aF # O for some a E U. Then define T: A -t 24"' by
T ( x ) = IF (= F I ) .
4.3. THE ALGEBRA 2l % 58
T is a non-zero continuous 2Lbimodule morphism from A into an**. There-
fore condition 3 in Theorem 4.12 does not hold for m = 1. Thus 31 @ A is
not 3-weakly amenable. O
Concerning question 1.5 in Chapter 1, Theorem 4.20 and Proposition 4.22
suggest that we might be able to find a counter-example in the Banach
algebras of the form 12L @ A. Unfortunately, B ( X ) can not be a candidate.
We will see this after the next two lemmas. The following lemma is basically
in [37, Theorem 81, but we have highlighted some of its features which are
important for our purposes.
LEMMA 4.23. Suppose that U is an infinite dimensional Hilbert space.
There are two elements Qo and So in B(X) such that for euch B E B(3L)
there exist PB E B ( X ) and RB E B(31) such thut ( (PB(( 5 IIBII, l lR~(l I 11B11
and
PROOF. First for an infinite dimensional Hilbert space 2, there exists
an isometry 1): 31 + CEP=, +xi, where C& + denotes the l2 direct sum, and
each Ri is a copy of 3C.
4.3- THE ALGEBRA 2f @ Zl 5 9
Let Q: 3t -t CEl +Xi and S: C& +Ri + xzl +?Li be the bounded
operators given by the infinite matrices
and S =
0 0 0
I O 0
C I O
0 0 1
* .
: : O
Let Qo = 8-1 O Q, So = q-l O S O q. Shen Qo, So E B(3L). For any element
B E B(31), let P: CE, +xi -t 3L and R: Cim_,-& -+ CZl +xi be the
bounded operators aven by the infinite matrices
P = ( B O O --) and R =
The following result on the 2nth d u d of B ( X ) seems not to be previously
known.
4.3. THE ALGEBRA II Zi
LEMMA 4.24. For any znteger n > 0,
B(%)(~") = span{au - ua; a E B (X), u E 8 (31)!*")) -
PROOF. By taking weak* limits and using induction, irnmediately the
result follows from the preceding lemma. O
PROPOSITION 4.25. For any m 3 0, B(31) @ B(Z) is (2m + 1)-weakly
amenable but is not 2m-weakly amenable.
PROOF. First as a C*-algebra B(U) is permanently weakly amenable. So
conditions 1 and 2 of Theorem 4.12 hold for X = U = B(R) and m 2 O. To
show that conditions 3 and 4 also hold it suffices to prove that any continuous
B(31)-bimodule morphism T from B(3t) into B(Z)("+') is trivial.
In fact, letting e be the identity of B(U) and F = T ( e ) , we have
T ( a ) = aF = F a for any a E B(3t). Therefore for any u E B ( X ) ( ~ ~ ) ,
(au - ua , Fj = O. From Lemma 4.24, this implies that F = O. Hence T = 0.
The above proves that B ( N ) @ B(R) is (2m + 1)-weakly amenable for
m 2 O.
Now we prove that condition 4 of Theorem 4.13 does not hold for m 3 O.
From Lemma 4.19, when X = B(Z), any B(7-l)-bimodule morphism T: X -t x ( ~ ~ )
satisfying condition 4 of Theorem 4.13 must be trivial. But the inclusion
mapping from X into x(~*) is a non-trivial B(3L)- bimodule morphism. This
shows that condition 4 of Theorem 4.13 does not hold. So B(31) 8 B(U) is
not 2m-weakly amenable for any m 2 O. The proof is complete. O
4.4. THE ALGEBItA S1@ Xo 6 1
Remark 4.26. Denote by K(3L) the compact operators on R. Using
Theorem 1 of [59] one can also prove that K ( X ) 8 K(R) and B(X) @ K(U)
are (2m + 1)( but not 2m)-weakly amenable. With this list: it is interesting
to recall Proposition 4.15 to see that K(31) 8 B(X) is not weakly amenable.
4.4. The algebra U @ Xo
In this section we consider the case that one sided (precisely, right) mod-
ule action on X is trivial. We denote this kind of 2l-bimodules specifically by
Xo. First we observe that when X = Xo conditions 3 and 4 in Theorem 4.12
are reduced, respectiveIy, to
3;. For any continuous Il-bimodule morphism r: Xo + there is
(2mf') such that Fa = O for a E II and r(z) = sF for z E Xo; F E Xo
4;. IIXo is dense in Xo.
PROOF. The equivalence of 3 and 3; in this case is clear. If 4 holds for
X = Xo, then Proposition 4.15 says that 21Xo (= %Xo + Xo%) is dense in
Xo. So 4; holds. Converseiy, if 4; holds, then any U-bimodule morphism
T: Xo + Xo (2m") is trivial, since the left %module action on X, (2m+1) iS
trivial. Therefore 4 holds. O
Also conditions 1, 3 and 4 in Theorem 4.13 are reduced, respectively, to
1; . For any continuous derivation D: +- if there is a continuous
operator T: Xo -t such that T(ax) = D(a)z + aT(z) for a E II
and x E Xo, then D is inner;
4.4. THE ALGEBRA II Xo 62
3; . Any continuous Z-bimodule morphism r: Xo + satisGing
r(z)y = O in xi2*) is trivial;
4: . For any continuous Il-bimodule morphisrn T: & + xizrn), there
exists u E such that aa = ua for a E U and T ( x ) = ux for
Suppose now that 'U has a b.a.i., then fiom C o r o l l q 3.7, condition 2 in
Theorem 4.12 always holds for X = Xo. This verifies the following theorem.
THEOREM 4.27. Suppose that 2l is a (2m + 1)-weakly amenable Banach
algebra with a b. a.i.. Then Q Xo zs (2712 + 1) -weakly amenable if and only
if conditions 3; and 4; hold.
COROLLARY 4.28. Under the assumptions of Theorem 4.27, if in addi-
t ion = Il(2m), then % @ Xo is (2772 + 1) -weakly amenable if and only
if rUXo zs dense in Xo.
P ROOF. 1f = 21(2m), then there is no non-zero continuous Il-bimodule
morphism T: Xo -t since
(au, T ( z ) ) = (u, T ( m ) ) = O (a E U,u E
So condition 3; holds automatically.
Especially for rn = 0, the above corollary gives:
4.4. THE ALGEBRA % @ -Yo 63
COROLLARY 4.29. Suppose that B is a weakly amenable Banach algebra
with a b. ai . . Then % @ Xo is weakly amenable if and only if UXo Zs dense
in Xo.
A dual result to Corollary 4.29 is as follows.
COROLLARY 4.30. Suppose that 2l Zs a wealily amenable Banach algebra
with a b.a.i., and fl zs a Banach Z-birnodule with the left module action
trivial. Then II @ fi is weakly arnenable i f and only if JZi zs dense in J.
The next theorem concerns the question: when weak arnenability implies
(2772 + 1)-weak arnenability?
THEOREM 4.31. Suppose that B is a n ideal of Y. Then the following
staternents are equiualent.
1. II @ Xo is (am + 1) -wea& amenable (rn 2 1);
2. II @ Xo is weakly amenable and ?L'(Il, = {O).
PROOF. It suffices to prove that 2 implies 1. If II@& is weakly amenable,
then Il is weakly amenable and 3O as well as 4; hold. From Theorem 3.5, U
is (2772 + 1)-weakly amenable.
Let r: Xo -t be a continuous 2Lbimodule morphism. Notice that
2f(*"+') = %*+IIL as 2bbimodules. Let P: + !ZL be the projection
with the kernel a*. Then P is an U-bimodule morphism. P O î = O since
4.4. THE .UGEBFU Q CB Xo
far as E UXO and u E
Here we have used the condition that 24 is an ideal of U'* and Lemma 3.3. So
I' is actually an a-bimodule morphism from Xo into II* and hence is O due to
Remark 4.14. This shows that 3; holds for al1 m. Hence from Theorem 4.12,
U @ Xo is (2m + 1)-weakly amenable if Yi1(%, xfrnf')) = {O). O
Remark 4.32. Theorem 4.31 can be viewed as an extension of Theo-
rem 3.5 in the sense that Theorem 3.5 deals with the case Xo = {O).
From Corollary 3.7, Theorem 4.31 yields immediately the following.
COROLLARY 4.33. Let lm 2 1 and suppose that U is a n ideal of II** and
has a b.a.i.. Then % @ Xo is (2m + 1)-weakly amenable if and only if it is
weakly amenable.
View U as a left %-module and then impose a trivial right 24-module
action on it. This results in a Banach U-bimodule. We denote it by Ao.
Similar to Lemma 4.19, one can check that the following equalities hold
- F# = O, u F = O, (Fu)" = Fu,
(2m+l) where u E u ( ' ~ ) , 4 E A!~), F E A. (m 1 0)-
4.4- THE ALGEBRA 2î @ Xo 65
PROPOSITION 4.34. Let m 2 O and suppose that 2i zs a (2m + 1)-weakly
amenable Banach algebra urith a o.ami.. Then II @ A. is (2m + 1)-weakly
amenable.
PROOF. Condition 4; holds automatically. R o m Theorem 4.27, it suf-
fices to show that condition 3; holds. In fact, let (x,) c A. such that (5,)
is a b a i . for U. Then for any x E Ao, we have x = IimZx,. Let F be a
(2m+Z) weak* cluster point of (I'(x,)), where F E A. , then
On the other hand
So condition 3; is fulfilled. The proof is complete.
Concerning 2m-weak amenability, we have the following.
PROPOSITION 4.35. Let m 3 I and suppose that U is a commutative
2m-weakly amenable Banach algebra with a b.a.i.. Then <2L @ A. is also
2m-weakly amenable.
PROOF. It suffices to prove that conditions 3; and 4: hold. Suppose
that an a-birnodule morphism r: A. -t satisfies r ( x ) y = O in AP), x, y E do. Then
4.4. THE ALGEBRA % @ Xo 66
This implies that r ( a ~ ) = O for a E Il, x E Ao, and so î(x) = O for al1
x E A. (notice that UAo = Ao). Therefore l? = O. Condition 3: holds.
For checking condition 4:, assume that T: A. + A O ~ ) is a continuous
Il-bimodule morphism. Let v be a weak* cluster point of ( T ( x i ) ) , where (3Fi)
is a b.a.i. for 2, and Iet u = S. Then
Hence T ( x ) = ux. On the other hand, ua = au since 24 is commutative. The
proof is complete. CI
Remark 4.36. In the preceding proposition, if 24 does not have an iden-
tity, then U @ A. is not O-weakly amenable. In fact, in this case 4; does not
hold for m = O, because for the identity mapping z: A. + Ao, the existence
of u E TU such that au = ua for a E U and z(x) = ux for x E A. means that li
has an identity u. So if B is a commutative weakly amenable Banach algebra
with a b.a.i. but with no identity, then % @ A. is a permanently weakly
amenable Banach algebra without being O-weakly amenable. Another known
example with this property is L1(G) with G an infinite, compact, non-abelian
group (see [17]).
4.4. THE ALGEBRA II @ Xo 67
Although we already have an example of a Banach algebra which is
(2m + 1)-weakly amenable but no t 2m-weakly amenable (Proposition 4.25;
another known example is the nuclear algebra N(E) with E a reflexive Ba-
nach space having the approximation property [17, Corollary 5-41), we end
this section by giving one more example of a weakly amenable Banach alge-
bra which is not 2-weakly amenable.
Suppose that U is a weakly amenable Banach algebra with a b.a.i. and
satisfying that U1(' # %*a (an example is Q = L1(G) with G a non-SIN
locally compact group; see (581 and [55] for the reference of SIN groups, and
[43, Theorem 32.441 as well as [56] for the property we need). Without loss
of generality assume UD* i a*%.
EXAMPLE 4.37. For the aboue Banach algebra Q, %@Ao is weaQ amenable
but is not 2-weakly amenable.
PROOF. From Proposition 4.34, A. is weakly amenable. We show
that condition 3; does not hold for m = 1. Take 4 E II*' sa t i seng #Imm. = O
but #lava # O (notice that, from Theorem 1.1, UU* is closed in a*). Then
$a = O for al1 a E B and ad # O for some a E 2.l. Let T: A. + a** be
defined by
4.4. THE ALGEBRA I11@ Xa 68
It is easily verified that T is a continuous U-bimodule morphism and T # 0.
Since
we have T ( x ) y = O for al1 x, y E Ao. But T # 0, and so condition 38 is not
satisfied for m = 1. Consequently S1 is not 2-weakly amenable. Cl
CHAPTER 5
Weak Amenability does not imply 3-Weak Amenability
In this chapter we answer the open question: Does weak amenability
imply bweak amenability? We first investigate the conditions of the weak
amenability for Banach algebras of the form II 8 (Xl+x2). Then we give
an example of a weakly amenable Banach algebra of this form which is not
3-weakly amenable.
5.1. Weak amenability of the algebra 3 @ (Xl+X2)
Suppose that X1 and X2 are two Banach Il-bimodules. We denote by
Xl+X2 the direct module surn of XI and X2, i.e. the l1 direct sum of Xl
and X2 with the module actions given by
For this module it is not hard to check the following equality:
In this section we study the weak amenability of the module extension
Banach algebra
5.1. WEAK AMENABILITY OF THE ALGEBRA 2l €EI (X l+Xz ) 70
PROPOSITION 5.1. Suppose that %@Xi and U@X2 are weakly amenable.
Then the following are equiualent.
(i) 2i @ (X1+X2) is weakly amenable;
(ii) there is no non-zero contznuous Il-bimodule morphisrn y: XI + X;;
(iii) there is no non-zero continuous Il-bimodule morphism q: X2 + X;.
PROOF. We first prove that (i) implies (ii). Assume that (Xl+X2)
is weakly amenable. Suppose that y: Xl + X; is a continuous Il-bimodule
morphism. Let T: xl+X2 i (XI+X2)* be the continuous U-bimodule
morphism forrnulated by
Then for (XI, xz) , (yl, y2) E Xl+X2, and a E a,
So (XI, x2) T((yl, y2)) + T((xl, x2)) (y1, y2) = O. Then from condition 4 of
Theorem 4.12, T = O. Thus y = O, showing that (ii) holds.
To prove that (ii) implies (iii), suppose that q: X2 + X; is a continu-
ous %-bimodule morphism. Then y: Xl + X; defined by y = q*lxi is a
5.1. WEAK AMENABILITY OF THE ALGEBRA Il ( x I + x ~ ) 71
continuous a-bimodule morphism. Therefore y = O, leading to that 7' = O
since q* is wk*-wk* continuous and Xi is weak* dense in X;'. Thus 7 = 0,
showing that (iii) holds. Similarly, one c m prove that (iii) implies (ii).
Finally, we prove that (ii) + (iii) implies (i) . Because B @ Xl and Ic e &
are weakly amenable, condition 1-3 of Theorem 4.12 hold automatically for
X = X1+X2 and m = O. So we only need to show that condition 4 also
holds. Suppose that T: X -+ X* is a continuous Il-bimodule morphism
satisfymg
Let Pi: X* -+ Xi' be the natural projections and 5: Xi -t X be the natural
embeddings, i = 1,2. Shen, by taking x2 = y2 = O and xl = yl = O
separately, we have
for al1 Xi, yi E Xi, i = 1,2. So we have Pi O T 0 ri = O, by applying con-
dition 4 of Theorem 4.12 to the weakly amenable Banach algebras II @ Xi,
i = 1,2. Furthemore, (ii) and (iii) imply that f i o T O r2: X2 + X; and
P2 O T 0 ri: Xi + X; are trivial. These show that T = O, i.e. condition 4 of
Theorem 4.12 holds. Consequently (i) is true. This completes the proof- Ci
5.2. WAK AMENABILITY 1 . NOT 3-WEAK AMENABILlTY 72
COROLLARY 5.2. U @ (X1+X2) is weakly amenable if and only if both
2i @ -XI and U @ X2 are weakly amenable and one of the conditions (ii) and
(iei) in Proposition 5.1 holds.
PROOF. If IL @ (&+x2) is w d d j amenable, then conditions 1-4 of
Theorem 4.12 hold for this algebra. It folIows that these conditions also
hold for the aigebras Xi and X2. SO they are also weakly amenable.
The rest is clear from Proposition 5.1. O
5.2. Weak amenability is not 3-weak amenability
Now we can construct an example of a weakly amenable Banach alge-
bra which is not 3-weakly amenable. In this section 3.1 denotes an infinite
dimensional separable Hilbert space. B(3L) denotes the Banach algebra of
all the bounded operators on X and K ( X ) the ideal of all the compact op-
erators on 2. It is well known that with Arens product as multiplication,
K(7Y)" = B(3F) ( see [57, pp. 1031 for details).
LEMMA 5.3. There is an element a0 E B(31) svch that a0 $ K(3C) , a0 is
not right invertible in B(U) and aoK(U) is dense in K(31).
PROOF. Let (e i )za=, be an orthonormal basis of 3t, and let a0 E B(R) be
defined by
(ei if i is odd.
5.2. WEAK AMENABILITY IS MOT 3-WEAK AMENABILiTY 73
Clearly a. @ K('H). Also a. is neither right nor left invertible because any
one-sided inverse of a0 must satisfy
ici if i is even; ai1[ei = (
ei if i is odd,
which can not be a bounded operator.
For any n 1 1, denote by Vn the subspace of 31 generated by {el, e?, . . . en),
and let Pn be the orthogonal projection from 3t to I;,. Then from Corol-
lary 11.4.5 of [14], for any k E K ( R ) and E 2 O, there is n = n(k, E ) , such
that
For this n = n ( k , E ) , let bn E B(31) be defined by
iei if i 5 n and i is even;
ei if i < n and i is odd;
O for i 2 n.
Then a00 bn = P, and a0 obno P, = P: = Pn. Let kn =brio Pno k. Shen
kn E K ( X ) , and a. O kn = Pn 0 li. A~SO
Since k E K(7f ) and E 3 O are arbitrarily given, this shows that aoK('fl) is
dense in K(31). The proof is complete. Cl
5.2. WEAK AMENABILITY IS NOT 3-WEAK AMENABILITY 74
For the elernent a. in the above lemma, aoB(3C) is a proper right ideal of
B(31) since the identity 1 $ aoB('H). The closure of a. B ( R ) is also a proper
right ideal of B(R) ([4, pp. 461). So there is F E B(X)* such that F # O
but Fao = O. Then FB(31) # {O) is a right B(X)-submodule of B(31)'.
Take
Then we have the following example.
EXAMPLE 5.4. B(X) @ (Xo+a ) is weakly amenable but not 3-weakly
amena ble.
PROOF. Clearly we have B(3C) Xo = Xo and f l B ( X ) = & By corol-
laries 4.29 and 4.30 each of the Banach algebras B(X) 3 Xo and B(X) @ $
is weakly amenable.
Suppose that T: a + X,' is a continuous B(7-L)-bimodule morphism.
We prove that T is trivial. Let f = T ( F ) . Then
and so
Therefore f = O since aoK(X) is dense in K(3t). This shows that T ( F ) = O
and hence T ( F B ( X ) ) = (O). Thus T = O. From Corollary 5.2, B ( X ) (Xo+ fi)
5.2. WL4K AMENABILITY IS NOT 3-WEAK AMENABILITY 75
is weakly amenable. Next we show that B(31) @ (Xo+ a) fails condition 4
of Theorem 4.12 for rn = 1. Since
we see that there exists a non-zero B(H)-bimodule morphism from a into
(XG)*** ( e g . the inclusion mapping). Let T : $! -+ (Xo)*** be such a mor-
phism, and let A: (Xo)*** -t (Xo)* be the projection with the kernel X:.
By taking T = A o r : OY -t Xi, and using the result of the preceding para-
graph, we have that T = O. So we have
Now let r: x0+ 8 -+ (Xo+ $)"* be the continuous B(X)-bimodule mor-
phism defined by
Here we used the fact u . xl = uxl E Xo (see Remark 3.4). So
5.2- WE-4E; APvIENABEITY IS NOT 3-WEAK AMENABILITY 76
for aL1 ( 2 1 , YI), (zz, M) E Xo+& But I' # O, showing that condition 4 of
Theorem 4.12 does not hold for m = 1. Consequently, B(3t) @ (&+a) is
not 3-weakly amenable. O
CHAPTER 6
Contractible and Reflexive Amenable Banach Algebras
It is a simple fact that any finite dimensional semi-simple Banach algebra
is contractible and, of course, (reflexive) amenable (see Assertion 1-3-68 and
Theorem VII.1.74 in 1391; for the definition of semi-simple Banach algebra
see Definition 6.8 on page 83). The converse is a challenging open problem.
We have mentioned the related references and made a brief comment on the
known results concerning this problem in Chapter 1. This chapter includes
two parts. In Section 1 we explore Taylor's method developed in [67]. In
Section 2 we treat the above problem from the viewpoint of some properties
of maximal ideals.
6.1. Approximation property and contractible Banach algebras
Let us first recall some basic concepts of functional analysis.
DEFINITION 6.1. A Banach space X is said to have the (compact) ap-
proximat ion property, AP (resp. CAP) in short, if for each compact sot
K E X and each E > O there is a finite rank (resp. compact) continuous
operator T on X so that IITx - 211 < E for al1 x E K. If T can be chosen so
that IlTl1 5 M for some constant M > O independent of K and E , then X 77
6.1. AP ALW CONTRACTIBLE ALGEBEMS 78
is said to have the bounded (compact) approximation property, BAP (resp.
BCAP) in short.
Briefly speaking, X has the AP if the identity operator on it can be
uniformly approximated on every compact subset by h i t e rank continuous
operators, and it has the CAP if can be uniformly approximated on every
compact subset by compact operators. If the n o m of the operators can be
chosen to be bounded by a fixed positive nurnber, then X has the BAP (resp.
BCAP). In Chapter 7 we will discuss more aspects of the approximation
property. Here we just point out that al1 Banach spaces with a basis enjoy
the BA? and the only naturally occuring Banach space known to lack this
property is B(N) with 3t an infinite dimensional Hilbert space. We refer to
[53] and [57] for more details about these concepts.
It is not hard to see that X has the BAP (BCAP) if and only if there exists
a number M > O and a net ((ox) of finite rank (resp. compact) operators on
X such that llpA 11 < M for a11 X and <px (x) + x for al1 x E X.
Suppose that X and Y are Banach spaces. We denote the projective
tensor product of X and Y by X h Y , and denote the elementary tensor of
x E X and y E X by x@ y; nre refer to [63] for the details about this kind of
product space. Suppose that 2l is a Banach algebra. Then UhU is naturally
a Banach Z-bimodule with the module multiplications specified by
6.1. AP AND CONTRACTIBLE ALGEBRAS 79
We use T to denote the bounded U-bimodule morphism from 118% into U
specified by a(a @ b) = ab, a , b E U.
DEFINITION 6.2. An element m E 0&2t is a diagonal for 5X if for al1
a E U, a rn = m - a , and a(m) is an identity of P. A bounded net (ma) of
Uh'U is an approximate diagonal for U if am, - m,a + O and ~ ( r n , ) a + a
in norm for al1 a E U.
It is a well-known fact that U is contractible if and only if there is a
diagonal for it [39, Assertion VII.1.721, and U is amenable if and only if
there is an approximate diagonal for it [47, Theorem 1-31. From these results
we notice that a contractible Banach algebra has dn identity, which will be
denoted by 1, and an amenable Banach algebra has a b.a.i..
Taylor proved in [67] that a contractible Banach algebra with the BAP
is finite dimensional. His method can actually yield more as shown in the
fol1owing theorem.
THEOREM 6.3. Suppose that 9i i s a contractible Banach algebra. If
has the BCAP, then it is finite dimensional.
PROOF. Suppose that u = CE, ai@bi is a diagonal for 2, where ail bi E U
satis& Cgl Ilaillllbill < m. Suppose that {(pA): X E A} is a net of compact
operators on TU such that Ilvxll < M, A E A, for some M > 0, and ipx(x) + x
6.1. AP AND CONTMCTIBLE ALGEBRAS
for all x E X. For each X E A, define Qx: U + U by
Then <Px is a bounded operator on B. Denote by 4x,n the nth partial sum of
the series in (6.1); i-e.
Then #A,, is a compact operator on U. For fixed X we have
in the uniform topology of operators. This shows that is compact. For
any a E St, we have
Here the first equality holds because the series converges uniformly. So GA
converges pointwise to 12, the identity mapping on Z. On the other hand
the bilinear mapping (a , b) H a<px(b) from 2l x IU into U defines a bounded
operator KPA: ~&?l -t Z satisfjing I A ( a 8 b) = apA(b). We have
Since a u = u . a we have for any a E U that
6.2. MAXIMAL IDEALS AND DIMENSIONS 8 1
It follows that converges to Ia in the uniform norm of operators (notice
that <U has an identity). This shows that 1% is compact. Consequently 2i is
finite dimensional. O
The same method can also prove the following result. We omit the proof
to avoid redundance.
PROPOSITION 6.4. Suppose that is a contractible Banach algebra. If
II has a diagonal u which has a finite sum representation u = x:., ai @ bi,
then 2i is finite dimensional.
Remark 6.5. It is known that amenable C*-algebras have the BAP
([IO]). So contractible C*-algebras are finite dimensional ( [65 ] ) .
Remark 6.6. A Banach space having a basis has the BAP. So a con-
tractible Banach algebra with the underlying space having a basis is finite
dimensional. Esp ecially, a contract ible Banach algebra with the underlying
space isomorphic to P(S ) (p 3 1) with S a non-empty set is finite dimen-
sional.
6.2. Maximal ideals and dimensions
If the underlying Baiiacl; space of ac amenable Banach algebra is reflex-
ive, we cal1 it a refEexzve Banach algebra. We note that both contractible
and reflexive amenable Banach algebras have an identity. In this section an
identity will simply be denoted by 1. Suppose that X, Y, Z are left (right)
6.2. MAXIMAL IDEALS AND DIMENSIONS 82
Banach %-modules and f: X -+ Y, g: Y + Z are continuous left (resp.
right) Il-module morphisrns. The short sequence
is exact if f is injective, g is surjective and I m ( f ) = Ker(g). The exact
sequence C is admissible if f has a continuous left inverse; and C splzts if
f has a continuous %-module morphism left inverse. We need the following
known result.
LEMMA 6.7 (Theorern 6.1 in [16]). Suppose that 2l is a contractible Ba-
nach algebra. Then every admissible short exact sequence of le0 or réght
Banach U-modules splifs.
We need also to recall some other general concepts. Suppose that B is
a Banach algebra. A proper left, right, or two-sided ideal of II is maximal
if there is no other proper left, right, or two-sided ideal of U containing it,
and is minimal if it contains no left, nght, or two-sided ideal of other than
{O) and itself. In a Banach algebra with an identity a maximal left, right,
or two-sided ideal is always closed.
For a E U, the limit
always exists [4, Proposition 2-81. r ( a ) is called the spectral radius of a. The
radical of an algebra is defined algebraically as the intersection of ail modular
6.2. MAXIMAL IDEALS .4ND DIMENSIONS 83
maximal left ide& (when the algebra has an identity, it is the intersection
of al1 maximal left ideals). However for Banach algebras we can adopt
DEFINITION 6.8. The radical of a Banach algebra % is
rad(%) = { q E II; r(aq) = O for al1 a E a).
If rad(%) = {O), then !U is said to be semi-simple.
A Banach algebra is called semi-prime if {O) is the only ideal J of it with
JZ = {O). The following basic facts are known:
(i) Semi-simple Banach algebras are semi-prime [4, Proposition 30.51;
(ii) For a Banach algebra 8, rad(%) is a closed ideal of 2l;
(iii) %/rad (a) is always semi-simple.
A division algebra is an algebra with an identity in which any non-zero
element has an inverse. It is known that a normed division algebra is always
finite dimemsional (see [4, Section 141). A non-zero element e of a Banach
algebra !Z is a minimal idempotent if e is an idempotent (Le. e2 = e) and eUe
is a division algebra. A well-known fact concerning minimal idempotents is
that for any two minimal idempotents el and e2 in IL, the dimension of ei%e2
is either O or 1, provided 2l is semi-prime [4, Theorem 31.61. There is a link
between minimal idempotents and minimal left (right) ideals described in
the following lemma.
6.2. MAXIMAL IDEALS AND DIMENSIONS 84
LEMMA 6.9 (Lemma 30.2 in [4]). Suppose that L is a minimal left (right)
ideal of <ZL such that L2 # { O ) . Then there exists an idempotent e E L such
that 2ie = L (resp. e<U = L); and each idempotent e vith Ue = L (resp.
e2l = L) is minimal.
For a subset E of M, the left annihilator and the right annilator of E are,
respectively, the following sets
l a n ( E ) = {a E I I ; a E = { O ) ) ,
r a n ( E ) = {a E I I ; Ea = { O ) ) .
For an element a E 2l, lan({a) ) and ran({a)) will be simply denoted by
lan(a) and ran(a) respectively. I t is easy to see the following.
LEMMA 6.10. Suppose that II is a Banach algebra having a n identity 1.
Then for any a E U,
(i) ran(U(1- a ) ) = ran(1 - a ) = {x E P; ax = x),
(ii) lan((1 - a)%) = lan(1 - a ) = {x E U; sa = x).
LEMMA 6.11. Suppose that 2l is a contractible or a refle2ive amenable
Banach algebra. Let L be a closed proper left (right) ideal of a. Ij L is
cornplemented in IU, then ran(L) (resp. lan(L)) contains an idempotent e
such that the following hold.
(i) ran(L) = e2i (resp. lan(L) = IIe);
(ii) L = P(1- e ) (resp. L = ( 1 - e ) a);
6.2. MAXIMAL IDEALS AND DIMENSIONS 85
(iii) If in addition, L is a maximal left (resp. right) ideal, then e is a
minimal idempotent and 2Ie (resp. e2l) is a minimal lefi (resp. right)
PROOF. We prove the case when L is a left ideal. So prove (i) and (ii)
we can assume L # {O), for otherwise e = 1 satisfies the requirements. First
we show that L contains a right identity p. Shen it is clear that p is an
idempotent and L = Up.
If B is contractible, we consider the following exact short sequence of left
U-modules:
where z is the inclusion mapping and q is the quotient mapping. Since L is
complemented, C is admissible. From Lemma 6.7 C is a splitting sequence,
Le. there is a left U-module morphism 6: II -+ L, such that 6 o 2 = IL, the
identity operator on L. Then p = 6(1) is obviously a right identity of L.
If 24 is reflexive and amenable, then according to Theorem 1.3 L contains
a right b.a.i., Say (1,). Since 24 is reflexive, as a closed subspace of a, L is
also reflexive. Then a weak* cluster point p of (1,) in L is a right identity
of L.
Now let e = 1 - p. Then e E ran(L), e # O and e is also an idempotent.
Since L = 2lp = % ( L e ) , from Lemma 6.10, ran(L) = {x E a; ex = x) = eM
This proves the first two staternents of this lemma.
6.2. b1AXIMAL IDEALS AND DIMENSIONS 86
Now suppose that L is a maximal left ideal. Then since U is the direct
sum of II(1 - e ) and %e we have that %e is a minimal left ideal of II. By
Lemma 6.9, e is a minimal idempotent.
LEMMA 6.12. Suppose that U i s a contmctible or a reflexive amenable
Banach algebra. IJ %/rad(%) zs finite dimensional, then so is U and U is
semi-simple.
PROOF. If TU/rad(ZL) is finite dimensional, then rad(%) is a comple-
mented closed ideal of %. From Lemma 6.11 rad(%) contains an idempotent
which is non-zero if rad(%) + {O). But rad(%) can never have a non-zero
idempotent from the definition. So rad(%) = { O ) . O
It is known that if a Banach algebra is contractible or amenable, then its
continuous algebra hornomorphic image is also contractible or amenable (see
[39, Proposition VII.1.711 and [49, Proposition 5-31). This leads to part of
the following Iemma.
LEMMA 6.13. Suppose that 2l i s a Banach algebra. Then the following
statements hold.
(i) If II is contractible, or repezi-ve and amenable, then so is %/rad(Q);
(ii) An ideal M c %/rad(%) is a maximal ideal if and only i f q - l ( M ) is a
maximal ideal in IU, where q: B -+ ILlrad(2l) i s the quotient mapping;
6.2. MAXIMAL IDEALS AND DIMENSIONS 81
(iii) If L is a maximal lefi (right) ideal o f% containing rad(%), then q ( L ) is
a maximal left (resp. right) ideal of U/rad(%). if L is complemented
in U, so is q ( L ) in %/rad(%).
PROOF. The first statement is clear. Checking of the second one is also
routine. Suppose that L is a left (right) ideal of 2l and rad(%) c L. Then
it is easily verified that L = q-'(q(L)) . So q ( L ) is maximal in %/rad(%)
whenever L is maximal in U. If L is complemented in U, then there is a
closed complement, Say J, of L in 'ZL. The image q ( J ) is also closed and
If m E q ( L ) n q ( J ) . then for some j E J and 1 E L , m = q ( j ) = q(1)-
Hence j - 1 E rad(%) c L. So j E L. This shows that j = O and hence
rn = O. Therefore q ( J ) is a cornplement of q ( L ) in %/rad(%). Thus q ( L ) is
complemented in Ulrad(2 i ) . 0
THEOREM 6.14. Suppose that 2l zs a contractible or a rejlexive amenable
Banach algebra. If each maximal ideal o f% is contained in either a maximal
left ideal or a maximal right ideal of 2i which is complernented in U, then 2i
is finite dimensional.
PROOF. F'rom Lemmas 6.12 and 6.13 we can assume that U is semi-
simple. We can also assume that is not a division algebra. Then Z
contains at least one maximal ideal and hence has at least one maximal
6.2. MAXIMAL IDEALS A . i i DIMENSIONS 88
left or maximal right ideal which is complemented in a. By Lemma 6.11,
II has at least one minimal idempotent. Let E be the set of al1 minimal
idempotents of Ur and let J be the ideal generated by E (called the socle of
a). We prove J = a.
If J # U, then there is a maximal ideal M containing J , since % has
an identity. Then by assumption, there is either a maximal left ideal or
a maximal right ideal of U which contains M and is complemented in II.
Assume the former is true and the corresponding left ideal is L. Then from
Lemma 6.11, ran(L) # {O) and contains a minimal idempotent e such that
L = U(l - e). So we would have Ee = {O) and e E E. This implies that
e = e2 = O, a contradiction.
Therefore J = U. Then the identity 1 of U can be expressed as
where ei, i = 1,2, ..., n, are minimal idempotents, and ai, bi E 2l. We then
have
Since each space eiIIej has a dimension of at most one, each subspace
ai(ei2kj)bj has a dimension of at most one. It follows that P is finite dimen-
sional. This completes the proof.
6.2. MAXIMAL IDEALS AND DIMENSIONS 89
Recall that a simple algebra is an algebra which has no proper ideals
other than the zero ideal.
COROLLARY 6.15. Suppose that Z i is a contractible or a rejlexive amenable
Banach algebra. If % is also a simple algebra and has a maximal le/t or right
ideal which i s complemented in Q, then 2i has a finite dimension.
PROOF. In this case, (0) is the only maximal ideal and is contained in a
maximal left or right ideal which is complemented in U by the assumption.
COROLLARY 6.16. Suppose that 2i is a contractible or a refiexzve amenable
Banach algebra. If every maximal ideal in B is either a maxzmal left or a
maximal right ideal, then Z i has a finite dimension.
PROOF. For each maximal ideal M, Z / M is a simple Banach algebra.
Since M itself is a maximal left or maximal right ideal in U, U/M has either
no non-zero proper left ideals or no non-zero proper right ideals, meaning
that {O) is either a maximal left or a maximal right ideal in %/M which
is certainly complemented. From the preceding corollary, U / M is of finite
dimension. Then any subspace containing M is finite codirnensional and
hence complemented in U. This is true for every maximal ideal M and so,
from Theorem 6.14, U is finite dimensional. O
COROLLARY 6.17 (Johnson [44]). Suppose that Q is a refiexzve amenable
Banach algebra. If every maximal lefl ideul of U i s complemented in U, t hen
6.2. MAXIMAL IDEALS AND DIMENSIONS 90
Qi is finite dimensional. In particdar, an amenable Banach algebra with the
underlying space a Hilbert space is finite dimensional (Ghahramani, Loy and
Willis [26]) .
PROOF. Since Q hm an identity, every maximal ideal of 2l is contained
in a maximal left ideal of P. Hence by Theorem 6.14, % is finite dimensional.
O
COROLLARY 6.18 (Runde [61]). Suppose that ZL is a repezive amenable
Banach algebra. If for euerg maximal ideal M , I(/M has a finfinite dimension,
then U has a finite dimension.
PROOF. As pointed out in the proof of the preceding corollary, each
maximal ideal M is contained i n a maximal left ideal N of II. Since UIM
has a finite dimension, N is finite codimensional and hence complemented
in U. From Theorem 6.14 we then have the result. O
CHAPTER 7
Approximate Complementability and Nilpotent Ideals
An ideal ,v in a Banach algebra is called nilpotent if there is an integer m
such that Nm = {O), where Nm = (nlnz - - -nm; ni E N,z = 172 , - - ,m l .
It is known that non-zero nilpotent ideals in amenable Banach algebras, if
any, must be infinite dimensional [16], and there do exist amenable Banach
algebras which contain non-zero nilpotent ideals (see [54] and [68]) . These
properties lead to the following natural question: Can a contractible Banach
algebra contain a non-zero nilpotent ideal? We see that this is impossible
if U has the BCAP since for this case we have shown in Chapter 6 that
2l is a finite dimensional (amenable) algebra (Theorem 6.3). In [54] Loy
and Willis have proved that if a biprojective Banach algebra has a central
approximate identity, then no non-zero nilpotent ideal can have the AP. In
Section 7.1 of this chapter we introduce the concept of approximately corn-
plemented subspaces and then discuss the relationship between this concept
and the known ones such as weak complementabitity and the approximation
property. In Section 7.2 we introduce a new class of Banach algebras called
the approximately biprojective Banach algebras. Then in Section 7.3 we will
improve Loy7s and Willis' result above by showing that a non-zero nilpotent
ideal in an approximately biprojective Banach algebra with both left and
7.1. APPROXIMATELY COMPLEMENTED SUBSPACES 92
right approximate identities can not be approximately complemented. We
will use some of the techniques of [54].
7.1. Approximately Complemented Subspaces
We first introduce the following:
DEFINITION 7.1. Suppose that E is a subspace of a normed space X.
Then E is approximately complemented in X if for each compact subset K
of E and each E > O, there is a continuous operator P : X -t E such that
Ilx - P(x)ll < E , for a11 x E K.
The following examples explain the use of the terminology "approxi-
mately complemented". First, if a subspace E of X is such that for every
compact subset K of E there is a subspace N, complemented in X, such
that K c N c E, then E is approximately complemented. In particular,
every complemented subspace is approximately complemented. A subspace
with the AP is also an approximately cornplemented subspace because every
finite ranlc bounded operator on the subspace can be extended to the whole
of the space. Another interesting example is that when E is a right (left)
ideal of a Banach algebra and has a left (resp. right) b.a.i., then E is ap-
proximately complemented. In fact, suppose that (e,) is a left b.a.i. of the
right ideal E. Then for each compact K c E and E > O, there is an index a!
such that 11%-e,xll < E for a11 x E K. So P : X + E defined by P ( x ) = e,x
satisfies the requirement in Definition 1. For C'-algebras, it is well known
7.1. APPROXIM-4TELY COMPLEMENTED SUBSPACES 93
that every closed left ideal has a right b.a.i. (see [50, Proposition 4-2-12]).
This shows that closed left ideds of a C*-algebra are approximately comple-
mented. For every Banach space X, let B(X) be the Banach algebra of al1
bounded operators on X with the multiplication of the operator composi-
tion, and let K ( X ) be the ideal of al1 compact operators on X. R o m [57,
Theorem 5.1.12(d)], K ( X ) has a left b.a.i. if X has the BCAP. So K(X)
is approximately complemented in B ( X ) if X has this property. It is worth
mentioning that K (X) is not complemented in B(X) when X is an infinite
dimensional Hilbert space (see [13]). For the theory concerning the existence
of left (right) b.a.i. in ideals of B (X) we refer to [21] and [62]. Also we
refer to [34] for concepts related to the approximate cornplementability we
just defined.
Suppose that X and Y are normed spaces. Denote by X ~ Y (reçp. X G Y )
the projective (resp. injective) tensor product of X and Y. We will use I I * I l x , Y
to denote the projective tensor norm in case any confusion may occur.
Suppose that XI, X2, YI and Y2 are normed spaces, and Ti: Xi -t Y;.,
i = 1,2, are bounded operators. We denote by Tl 8 T2: X&X* + Yi6Y2
the linear operator specified by
7.1. APPROXIMATELY COMPLEMENTED SUBSP.4CES
are bounded operator sequences of normed spaces, then we have
LEMMA 7.2. Suppose tha t X is a n o m e d space and E zs an approxi-
rnately complernented subspace of X. Let z : E + X be the inclusion map-
ping. Then 2 8 IF : E ~ Y + X&Y zs injective for every norrned space Y ,
where Iy iS the identi ty rnapping on Y .
PROOF. Let u E E&Y satisSing z 8 &(u) = O in X ~ Y . We prove
u = O in E ~ Y , or equivalently, I I u I I ~ , ~ = O. Tt is elementary that for every
convergent series Cr==, x,, x, > O, there exists a sequence (h) such that
c, -t oo and CF=l C n x n converges. Thus, without loss of generality, we can
assume u = CF=P=, an 8 bn with a, E E and b, E Y, n = 1,2 , ..., satisfying
lirn,,, a, = O, CE1 11 bn 11 < 1. Given E > O, let P : X -t E be a continuous
operator satis&ing Ilan - P(a,)II < E for a11 n. Then P B I y : X ~ Y -t E 6 Y
is a continuous operator and hence
Therefore
Since E was arbitrary, we have I I u I I ~ , ~ = O and hence z 8 Iy is injective. O
7.1. APPROXIMATELY COMPLEMENTED SUBSPACES 95
Remark 7.3. One can not expect that z 8 Iy will be injective for each
closed subspace E of X. In fact, if X is a Banach space with the approx-
imation property and E is a closed subspace of X without this property,
then there is a Banach space Y such that z 8 ly: E ~ Y + X ~ Y has a non-
trivial kernel. Because, (see [35, p.1561) if E does not have the approxima-
tion property, then there is a Banach space Y such that a non-zero element
u = CT=lan @ b, E E&Y exists with C;=!an @ b, = O in E 6 Y . Since
E6Y c X Q Y , xC1 an 8 bn = O in X6Y. This implies CCl a, 8 bn = O in
x ~ Y , since X has the approximation property. In other words z@Iy(u) = 0.
The problem of under whst condition z @ Iy is injective is still open.
COROLLARY 7.4. Suppose that Ti : Ai -+ Bi, i = 1,2, are topologically
isomorphic embeddzngs (injective and homeomorphic) of Banach spaces. If
Ti(Ai) is approximately complemented in Bi, i = 1,2, then the operator
Tl 8 T2 is injective.
PROOF. Let : Ai + T,(Ai) , i = 1,2, be the canonical isomorphisms
induced from z, and zi : z(Ai) -+ Bi, i = 1,2, be the inclusion mappings.
Then
The operators IB1 @ T~ and pl @ la, are invertible. Frorn the condition and
the preceding lemma, IB, 8 z2 and 21 8 IA, are injective. Therefore, as a
composition of injective operators Tl 8 T2 is injective. Cl
7.1. APPROXIMATELY COMPLEMENTED SUBSPACES
For Banach spaces with the AP we have the following result.
THEOREM 7.5. Suppose that X is a Banach space having the AP. T h e n
a subspace E is app~oximately complernented in X if and only if E has the
AP.
PROOF. We only need to prove the necessity. Suppose that E is approx-
imateIy complemented in X. Suppose that K is a compact subset of E.
Then for each E > O, there is a bounded operator Pl: X + E, such that
I I Pt(k) - kll < ~ / 2 , 1; E K. Also there is a bounded finite-rank operator P2
on X such that IIP2(k) - kll < E / ( ~ ~ ! P ~ I I ) . Let P = (Pl O P2)IE. Then P is a
bounded finite-rank operator on E.
for k E K. This shows that E has the AP. O
It is known that many Banach spaces enjoying the AP contain subspaces
which lack the AP ([24], [18] and [66]). From the above theorem, such
subspaces are not approximately complemented .
A subspace E of a Banach space X is said to be weakly complemented in
X if the dual operator z* of the inclusion mapping z: E + X has a continuous
right inverse ([40, p.231). In Banach spaces literature, this concept was k s t
discussed in [51], where the terrninology local complementability was used
for it. An equivalent, but more simple, expression of this concept is that
7.1. APPROXIMATELY COMPLEMENTED SUBSP-4CES 97
EL = { f E X*; f (x) = O for x E E) is complemented in X* (see [16] and
[41]). In fact, one can see this equivalence easily by noting that k e ~ ( z * ) = EL.
In the following we are concerned with the relationship between approx-
imate complementability and weak complementability. First the following
example shows that the latter does not cover the former.
Recall that an infinite dimensional Banach space is prime if it is iso-
morphic to every infinite dimensional complemented subspace of it (see [53,
Definition 2.a.61).
EXAMPLE 7.6. Consider the Banach space L1[O, 11. It has a subspace iso-
morphic to l2 053, p.571). This subspace is approximately complemented in
L1[O, 11 since lî has the approximation property. W e proue that it zs not com-
plemented and hence frorn Corollary 7.8 below it is not weakly complernented
since it is a daal space.
PROOF. Suppose, towards a contradiction, that this subspace is comple-
mented. Then Lm [O, 11, the conjugate of L1[O, 11, would have a complemented
subspace isomorphic to 1 2 . Since Lm[O, 11 is isomorphic to 1, ( [53, p. l l l ] ) ,
1, would have a complemented subspace which is isomorphic to 12. But Zoo
is a prime space ([53, p.571). This would imply that 1, were isomorphic to
12, which is impossible. O
It is not known whether every weakly complemented subspace is approxi-
mately cornplemented. In amenable Banach algebras, if the subspace is a left
7.1. APPROXIM,4TELY COMPLEMENTED SUBSPACES 98
(right) ideal, then this is true because it d l have a right (resp. left) b.a.i.
in this case (see Theorem 1.3). For the general case, to discuss this problem
the following characterizing result for weak complementability seems to be
useful.
PROPOSITION 7.7. Suppose that M i s a closed subspace of a Banach
space X . T h e n M is weakly complernented in X if and only i f there is a
bounded operator P: X + Al**, such that, for each m E M , P o z(m) = f i ,
where z: M -t X is the inclusion mapping and m zs the image of rn in M**
under the canonical mapping.
PROOF. To prove the necessity we assume that M is weakly comple-
mented and p: M* + X* is a continuous right inverse of 2'. Then p*: X** + M**
is a bounded operator satisfying p* O z** = IM-. Let j : X + X** be the
canonical mapping and set P = p* O j . Then we have
For proving the sufficiency, suppose that P: X + M*' is a continuous
operator such that P 0 z(m) = m for rn E M. Then p = P'I,M-: M* + X*
sat isfies
(7% z* op(m*)) = (m*, P 0 z(m)) = (m*, iîl) = (m, m*),
for each m* E M* and m E M. So z* o p(m*) = m* for al1 m* E M*, i-e.
z* 0 p = I lM. . Therefore M is weakly complemented.
7.1- APPROXliM-4TELY COMPLEMENTED SUBSPACES
From the preceding proposition we have the following.
COROLLARY 7.8. Suppose that M is a veakly complemented subspace of
a Banach space X . If M is also a dual space, then M is complernented in
X .
PROOF. Since M is a dual space, there is a bounded operator 0: M** +
M, such that a ( h ) = m for m E M , where fh is the image of m in M** under
the canonical mapping. Let P: X M** and z: M + X be the bounded
operators described in Proposition 7.7, and let PM = Q o P: X + M. Then
PM(z(m)) = m for rn E M. So 2 O PM is a bounded projection from X onto
z (M) . Therefore z(M) (i-e. M) is complemented in X. O
COROLLARY 7.9. Suppose that M is a weakly complemented subspace of
a Banach space X . If M is approximately complemented in M**, t h e n it is
approxirnately complemented in X .
PROOF. If M is approximately complemented in M**, then for every
compact set K c M and every E > O there is a bounded operator T: W* +
M , such that IIT(~) - kll < a for k E K. Let P and z be the operators
described in Proposition 7.7. Then S = z o T o P: X + 2(M) will satis&
IlS(z(k)) - z(k)ll = IIT(&) - kll < a for k E K. This shows that 2(M) (Le.
M) is approximately complemented in X. Cf
7.2. APPROWMATELY BIPROJECTIVE B.ANACH ALGEBHAS 100
7.2. Approximately Biprojective Banach Algebras
In this section we introduce the following notion.
DEFINITION 7.10. A Banach algebra TU is pointwise approximately bipro-
jective if for each a E II there is a net {Ta 1 a E A) of continuous bimodule
morphisms £rom U into such that a O T,(a) -t a in norm. If in addi-
tion, the net {Ta ( a E A) is independent of the element a E 2, then 5% is
approximately Oiprojective.
Of course every biprojective Banach algebra is approximately biprojec-
tive. Al.so, if II has a central approxirnate diagonal (Le. a net {u,1 a E A)
in U&U such that au, = u,a, for al1 a E II and a E A, and a(au,) + a
in norm), then II is approximately biprojective. An example of such kind of
Banach algebra is as follows.
EXAMPLE 7.11. Let S be any non-empty set and consider t2(S) with the
pointwise multiplication. Let A be the collection of al1 finite svbsets of S
ordered by inclusion. Then {u , = Ci,, ei @ ei 1 a E A) fonns a central
approximate diagonal of 12(S), where ei is the element of 12(S) equal to 1
at i and O elsewhere. So t2(s) is approximately biprojective. W e prove that
t2 (S) is not biprojective whenever S is a n infinite set.
PROOF. Suppose conversely that T : t 2 ( S ) -t 12(s)&?(S) were a con-
tinuous bimodule morphism satisfying TOT = Ip(s). Since T(e i ) = eiT(ei)ei,
7.3. NILPOTENT IDEALS 101
one sees easily that T(e i ) = ei @ ei for each i E S. SO if x = Cies aie; E 12(S),
then T ( x ) = Ces aiei@ei. Consider the identity operator z E BL(!~(S), e2(S)),
which can be viewed as an element of (P (S)&P (S))', when the latter is iden-
tified with BL(t2 (S), 12(S)) (cf. [4, 8301). We have
But it is clear that (ei 8 ei, 2) = 1 for each i E S, and so
Together with the preceding inequality, one is led to a contradiction that
Cies ai converges for each x = CiEs aiei E 12(S). O
Remark 7.12. The above example also shows that, for each commuta-
tive compact group G, L2(G) with convolution multiplication is approxi-
mately biprojective but is not biprojective unless G is finite, In fact, from
Plancherel's Theorem ([6O, Theorem 1.6-11) L2 (G) is isometrically isomor-
phic to f?(r); being the dual group of G and t 2 ( r ) having pointwise
multiplication. It is worth mentioning that, although L2(G) has a central
approximate diagonal as shown in the example, by Coroliary 6.17 it is not
amenable unless G is a finite group.
7.3. Nilpotent ideais
In the following for two subsets M and N of an algebra Z, MN always
denotes the set {mn 1 m E M , n E N ) .
7.3. NILPOTENT lDEALS 102
LEMMA 7.13. Suppose that II is a pointwise approximately biprojective
Banach algebra, N i s an approximately complemented closed ideal of I I , and
E is a closed ideal of II satisfying E C N and EN = { O ) (NE = {O)).
- - Then for every subset M of 2i satisfying M E C EII (resp. E M C ZE), it
is true that M E = {O) (resp. E M = {O)).
PROOF. We prove the case EN = {O) and ME C m. Let z : N + 2l
be the inclusion operator, q : II + 2i /E be the quotient operator, la, 1 , ~
and lalE be the identity operators on Tn, N and 2lIE respectively, and
p : (%/E)&N + N be the operator specified by p((a + E ) 8 c ) = ac for each
a + E E %/E and c E N. The operator p is well defined because EN = { O ) .
All the operators above are obviously continuous.
Suppose towards a contradiction that ME # {O), and assume mc # O
for some rn E M,c E E. From the assumption rnc = limGa,, for some
sequences (ÇJ c E and (a,) c U. Since 2l is pointwise approximately
biprojective, there is a continuous bimodule morphism T : Q + 0 6 2 i such
that n o T(m) c # 0. For b E N, let Rb (L6) : II + N be the map of right
(resp. left) multiplication by b. We now consider the element d E ( 2 i / ~ ) & N
defined by
7.3. NILPOTENT IDE.4LS
we have d f O. Now consider its image under laIE 8 2.
contradiction which shows that ME = {O).
THEOREM 7.14. Suppose that 2t is a pointwise approximately biprojec-
tive Banach algebra. If U has both left and right approximate identities,
then II possesses no non-zero nilpotent ideal whose closure is approximately
complernented in U.
PROOF. If there were a non-zero nilpotent ideal N whose closure is ap-
proximately complemented, then there would be a n integer n > 1 such
that N* # {O) and Nnfl = {O). Without loss of generality, asurne that
N is closed. Let E be the closure of the linear span of Nn. Then E
7.3. NILPOTENT IDEALS 104
would be a non-zero closed ideal of satiswng E c N and EN = {O).
Since U h a a left approximate identity, E = I1E. Similarly E = E2l. So
- U E EU. Then 111E = O from Lemma 2, which leads to a contradiction
that E = U E = {O). O
Remark 7.15. The requirement of the existence of both left and right
approximate identities in the above theorem can not be reduced to the exis-
tence of just one one-sided approximate identity, as seen from the following
example of Helemskii. The Banach algebra of 2 x 2 matrices with zero first
column is biprojective. It has a right identity (X 1) but no left (approximate)
identity, and it does have a non-zero nilpotent ideal (which must be comple-
mented since the algebra is finite dimensional) containing al1 the matrices of
the forrn ( a 8 ) .
COROLLARY 7.16. A pointwise approximately biprojective Banach alge-
bra with both left and right approximate identities has no non-zero finite
dimensional nilpotent ideals.
PROOF. Any finite dimensional subspace is complemented.
COROLLARY 7.17. Suppose that 2l is a pointwise approximately biprojec-
tive Banach algebra whose underlying space is a Hilbert space. If 2l has both
left and right approxirnate identities, then possesses no non-zero nilpotent
ideals. If in addition, 2l is commutativet then it has no nontrivial nilpotent
elements.
4.3. NILPOTENT IDEALS 105
PROOF. Every closed subspace of a Hilbert space is complemented. If in
addition, U is commutative, then the ideal generated by a nilpotent element
is nilpotent. Cl
Since contractible Banach algebras are biprojective and have identities,
we can deduce immediately the following result.
COROLLARY 7.18. Suppose that 2i is a contractible Banach algebra. Then
a possesses no non- zero ndpotent ideah which are approximately comple-
mented.
Similar results can be obtained for Banach algebras with central approx-
imate diagonals. For example we have:
COROLLARY 7.19. Suppose that II is a Banach algebra with a central
approxzmate diagonal. Then 2l possesses no non-zero nilpotent ideal which
is approximately complemented.
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