1C/93/405INTERNAL REPORT(Limited Distribution)

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON (^COMPLEMENTED NORMED ALGEBRAS

Etienne Desquith 'International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

Given a linear space X and a linear form F on it, there is up to isomorphism aunique way to endow this space with a left commutative algebra structure admitting Fas unique multiplicative linear form. This algebra is not commutative, but each of its onedimensional linear subspaces is a subalgebra (even a left ideal) naturally isomorphic tothe complex <T(cf.[3]; [4]).

It will be shown that linear forms can also be used to define commutative algebraspossessing no other multiplicative linear forms than those ones precisely used to definetheir products. Each such algebra satisfies the property that some subalgebra in it isisomorphic to W, for some n > 1. This motivates the study of general algebras possessingthe later property and called for this (^-complemented algebras.

MIRAMARE - TRIESTE

December 1993

'Permanent address: Institut de Recherches Mathematiques (IRMA), 08 BP 2030Abidjan 08, Ivory Coast.

' I-" r ~ - •

1 Introduction

Finite n-dimensional linear spaces are known to be linearly isomorphic toff", But forn > 1, and under coordinatewise multiplication, (T" is a commutative algebra, and then,it is no longer true that any finite n-dimensional algebra A is isomorphic to (Tn in thesense that there is an algebra morphismfrom A ontotF". For example the algebra Mn(ff)of n x n complex matrices is not isomorphic to ff" as an algebra.

The purpose of this paper is to show that given any Banach space X , it is alwayspossible to turn it into a normed algebra admitting an isomorphic image of <Z"\ Theproduct of such an algebra will be given via continuous linear forms on X and the resultingalgebra will be compact, will admit no multiplicative linear forms except those ones usedto define its product. It will also satisfy the Wedderburn property with a finite dimensionalsubalgebra for topological complement of its radical. Dealing with such a space naturallyraised our interest for general algebras having subalgebras isomorphic tod"1, the so-called(^''-complemented algebras.

Thoughout the sequel we shall use the following definitions:A sequence (u;),>! in an infinite dimensional Banach space X will be called a (Schauder)basis of X , if for x€ X there is a unique sequence of scalars (x,),>i such that

CO o&

x = 2_JX*U" or cquivalently : lim | | j — ^ i ; u , - | | = 0.

The continuous linear forms (F,-),>i on X satisfying Fi}(uj) = <5,j (where 6i} is theKronecker symbol) are the so-called "associated sequence of coefficient functionals"(a.s.c.f., for short). A character oi' a normed algebra A will be a continuous multiplicativelinear form on it, and A is said to satisfy the Wedderburn Property, provided it is thetopological direct sum of its radical and a closed subalgebra. The (Jacobson) radicalof A means the intersection of the kernels of all irreducible representations of A . Thetopological dual and bidual of a linear space X will respectively be denoted by X' and<Y". All linear spaces will be over the complex field d.

The work will be organized as follows: In Section two, we deal with normed algebrasdefined by a finite number of linear forms and with (T™-complemented algebras for finiten, while section three will be devoted to normed algebras <T"- complemented for all n > 1.

2 (^''-Complemented normed algebras for finite n

Given a complex linear space X and a linear form F on it, an algebra is obtained throughthe following product of any x , y € X :

xy = F(x)y. (1)

In ([4]), we have made a systematic study of the case where the space is the topologicaldual X' of a Banach space X , and F = (ic € X" is defined by

ltt(T) = T(e) (2)

where e is a nonzero element of the unit bait of X . We had then shown that the Banachalgebra (A"*,[e]) obtained satisfy the following properties:

(Pi) • (X",[eJ) is left commutative but not commutative when X is of dimension greaterthan one.(Pi) '• Each of its linear subspaces is a left ideal.(P3) : If F' is another continuous linear form of norm less than one on X', then the prod-ucts defined via F and F' are different, but they give rise to isomorphic Banach algebras.(P4) : (.Y*,[e]) admits a unique character, namely ftt , and its radical coincides with thekernel Ker(ttt) of fit. Hence it satisfies tlie Wedderburn Property.(Pi) : For any T E (X',\eJ) to be quasi-invertible, it is just sufficient that its spectralradius r(T) fulfils: r(T) # 1.

However, the concept of left commutativity is not a usual one. So we tried to know inwhich other ways linear forms could be used to construct commutative algebras. Quitesatisfactorily, this question can be answered positively through what follows.

N.B.: From now on, we shall denote the algebra defined on a linear space X by means ofa linear form F by (X; F) , and by (X; Flt • • •, Fn), the one defined via the linear formsFj, • • •, Fn. Moreover, although what follows applies to general normed linear spaces, weshall essentially deal with Banach spaces, and all linear forms will be assumed continuous.

Let F be a linear form on a Banach space X , and e 6 X such that F(e) ^ 0. ThenX becomes a complete compact commutative normed algebra (X;F) when endowed withthe following product:

xj, = F(x) F(y) e (3)

such that:(i) If F and e are of norm less than one, then (X; F) is a Banach algebra under theBanach space norm.(ii) If F(e) = 1 , then (X;F) admits e and F respectively as unique idempotent andunique character.(iii) (X;F) has Ker(F) for radical and therefore satisfies the Wedderburn Property.

Remark 2 .1 . If (A\(jF]) is the Banach algebra denned in (1), then both (A'jfjFj) and(X;F) admit the same radical (Ker(F)), but they are of quite different kinds:- (X;F) is commutative but (-Y,[Fj) is not.

- Every (Tx for x ^ 0 in X , is a subalgebra of (A\[fj) isomorphic to W, while We is theonly subalgebra of (X;F) isomorphic to(T.

Proposi t ion 2.2. Let {ei,e2} be a linearly independant set in a Banach space X ,and let Fx and F2 be the elements in X' such that Fi3(tj) = 6{l ; i,j = 1,2. Then theoperation o n A x X defined for all (x,y) G X x X by

xy = Fjfz) Fi(y) e, + F2(x) F2(y) e2, (4)

turns X into a commutative normed algebra (X; Fi,F2).

Proof. Straightforward. •

Let || || denote the Banach space norm of X . We have:

Proposi t ion 2.3. Let ei and e2 in proposition 2.2 be of norm less than one. Thenunder the new norm

*HWI = (ll^i|lJ + FaH2)H| , (5)X becomes a Banach algebra, with only two characters Fs and F2.

Proof. That F\ and F2 are nonzero multiplicative linear forms is immediate from relation(4). Let (ft bean arbitrary multiplicative linear form. Then since et and c2 are idi mpotentelements, we have:

Without loss of generality, we can assume that ^(e,) = 1. But since e,e2 = 0, tj> necessarilysatisfies: <J>(e2) = 0, and from

ie, = f , ( i ) e, ; x e (X; Flt Fa),

we get 4> ~ Fi- Assuming ^(e2) = 1 similarly gives 4> = F2.Let u2 = (HFill2 + 1|F3||

2), and define

Then for all x, y £ X, we get:

(7)

(8)

and the proof is complete. D

Proposi t ion 2.4. The set of ideals of of (X; FUF2) includes the following subspaces:ffej, (Te2, Ker(Fi), Ker(F2), and all linear subspace of X containing the two-dimensionalclosed linear subspace [ei,e2] generated by ei and ej.

Proof, Easy computations show thatu^ej and#e2 are ideals. Fl and F2 being characters,their respective kernels are also ideals. Now, if I is a linear subspace of X containing [ei, e2],then for each x € / and for all a e (X; Fi, F2), it comes:

ax = Fi(o) F2(i) Cl + F2(a)

that is, I is an ideal. •

(9)

If X is two-dimensional, then (X;FUF2) has a unit element u = e, + e2 , and isisomorphic to df1. The group G of invertibte elements then is:

G = /„, \{X; F,,F2)] = X\ {(Te, U(Te2} , (10)

and for each x e (Jf;F,,Fj), the spectrum <r(x) is given by: <r(x) = [l\(x), F2{x)}.

We have the following characterization of <Xn-complemented normed algebras.

P r o p o s i t i o n 2 . 5 . A normed algebra A is (F"-complemented if and only if it contains n

pairwise orthogonal idempotents (ei)i<i<n), that is:

e.-ej- = V ; ; 1 <i,j <n. (11)

P r o o f . The standard basis (e;)i<<<Ti of (Fn clearly satisfies relation (11). So if p is anisomorphism from Wn into A , then the subset {p{e.) • i — 1, • • • ,n} of A consists ofpairwise orthogonal idempotents of A . Conversely, if {«i, • • • ,uB} is a set of pairwiseorthogonal idempotents of A , then the UiS are linearly independant, Bn = \u\,- • • ,vn] isa subalgebra of A , and there exists a linear isomorphism 8 from Bn onto (Fn. Then thee; = $(ui)s give n linearly independant elements in(T" satisfying: e^ej = 6ijej , for alli — 1, • • • , n ; and for all a, b 6 Bn, 6 clearly satisfies

6(ab) = 8(a) 9(b) ,

and is therefore an algebra morphism. D

(12)

Example 2.6. The algebra M3(<T), of 3 x 3 complex matrices is (T3-compleinented.Indeed, the set {Fi,P2,P3} where

consists of pairwise orthogonal idempotents in Mj((F). Moreover, if M3{(E) is d"1-complemented, for some n > 3, then the subspace [Mi, M2, • - -, Mn] isomorphic to (F"(and composed of pairwise orthogonal idempotents) does not include {Pi, P2,Ps] , for if{Pi, P2, P3, M} is a set of pairwise orthogonal idempotents in M^tfE), then M is necessarilythe zero matrix.

More generally, if A is a Banach algebra with unit e = ^ " = i e; , where the e^ arepairwise orthogonai idempotents, then {ei, • • •, en} is a maxima} set of pair wise orthogo-nal idempotents.

Remark 2.7. A (^-complemented normed algebra contains a n-dimensional commu-tative subaigebra. Conversely, is each commutative n-dimensional normed algebra (En-complemented?

The answer is no, according to the following counter-example: the algebra (X; F) withproduct given by (3) is commutative and n-dimensional (n > 1), if n = dim(X) > 1 .But as seen before it is not (Hk- complemented for any k / 1.

It follows from Proposition 2.5 that a ^"-complemented normed algebra admits a biorthog-onal system (ei,fi)i<i<n such that (e;)i<i<n is a set of pairwise orthogonal idempotentsof A , (F,),<,<n C A', and /^(e,-) = S,j"We therefore set:

Definition 2.8. Given a normed algebra A , a biorthogonat system (e,, F;)i<i<n in which{ej is a set of pairwise orthogonal idempotents is called an idempotent biorthogonal sys-tem.

Hence to any (f-complemented normed algebra, is associated at least an idempotentbiorthogonal system.

Theorem 2.9. Let X be a Banach space and (e,-,Fi)i<,<n a biorthogonal system. Thenthe operation

X , (13)

defines a product on X , and the commutative complete normed algebra obtained is W"-complemented and admits {f\, Fj, • • •, FnJ as set of nonzero characters. Moreover, it canbe renormed in such a way that it becomes a Banach algebra.

P r o o f . That relation (13) defines an associative product on X , is checked by routinecomputations, and it is also easy to see that this product is commutative. Moreover,relation (13) makes immediate the fact that {ei, • • • ,e n} is a set of pairwise orthogonalidempotents of (X; Fi, • • •, Fn), which, according to Proposition 2.5 , is C" -complemented.Now let ^ be a character of (X; F\,-- • ,Fn). Then for all x,y G {X; Fj, • • • , F n ) , we get

4>{xy) = Y^Fi(x) Fi(y) 0(e,) = (14)

But since e, is idempotent for each i € {1,2, • • • , n } , the <j>(ti)s satisfy

M e , ) ;« = l , 2 , - - , n } C {0,1} (15)

Let j be chosen arbitrarily in {1,2, • • • , « } . Without loss of general i ty , we may assumethat <j>(e3) = 1. Then it follows

(16)

and taking x = e3 , y — tj + ft , 1 < h < n ; k ^ j , leads to:

n

1 + Y, F,(e]) F,(e3 + ek) <t>(et) = ] + 4>(e

which implies that4>{ck) = 0 , V I < A < n ; k£j

But then, for all x € {X\ F,, • • •, Fn), we have:

that is,

<j>(xe3) = 4{x) = Y. fiW F'^ ^<) = F&) <

<j> = Fj , for some j 6 {1,2, •• • ,ra}.

(18)

(19)

(20)

Theforere, the set of characters of (X; Fu • • •, Fn) coincides with {Fi, • • •, Fn}, each suchmap being obviously continous and hence a character. If we set

(21)

then a new norm on {A'; Fj, • • •, Fn) is defined as follows:

* H | * | | , n , = ^11*11; x e ( X ; F u - - - , F » ) , (22)and all x, y e (A; Fj, • • •, Fn) satisfy

< ^INIIMI< Nwbl lw

Hence, under the norm || ||(n), (X; Fj, • • •, F,,) is a Banach algebra (A'; Fi,•• •, Fn ; | |[in)}which witl be denoted in the remainding of the sequel by: (Fn)(JV). •

Coro l l a ry 2.10. Given a Banach space and n linearly independant linear forms on it,there exists a Batiach algebra structure on X for which the F;3 are the only characters.

Proof. It is sufficient, according to Theorem 2.9 , to show that there are elementscii e2, ••,«„ in X such that (e<, F;)i<,<n is a biorthogonal system, and then the desiredBanach algebra will be F^)(X). For each j g (1,2,- •- ,n} , it is sufficient to pick u ; inthe set

Ej= f) Ker(F,) \ KcriF,) , (23)

and consider z} = UJ/FJ(UJ}. D

R e m a r k 2 .11 . Every n-tuple of linear forms (#j)i<,<n such that the restriction of eachHj to [f], • • • ,£„], say Gj coincides with a unique Fj g {F\,- • • ,Fn}, defines the sameBanach algebra F(n)(X) as the F;s.

T h e o r e m 2.12. Under the hypotheses of Theorem 2.9 , FM(X) is a compact Banachalgebra not (Tk-complemented for any k > n.

Proof. Given z € A', the linear map x t-> z x z has finite m-dimensional range m < nand is then compact. Now, assume that F(n)(A') i= <Tn+k-complemented with k > 1, andlet a be an isomorphism from <Tn+': onto the subalgebra .4n+t = ["1, • • • ,«n+t] , where{«!,- • • ,un+t}, is the iniage of the canonical basis of (F™+*. If e is the unit element of<Fn+k, then

is the unit element of v4n+,t, and has the decomposition

u = y^ u, .=1

But relation (24) implies

(24)

(25)

(26)

It follows that

which, "since the ds are linearly independant, gives:

F,(u) = 0 , or 1 , V 1 < i < }> .

Hencen+i p

i=l i=l

But then, each u; G An+k fulfils:

(27)

(28)

(29)

u, = u tij = > e, uj

which forces the (n+k)-dimensional space /!„+* to be contained in the n-dimensional sub-space [ei, • • • ,**]• This being absurd, the assumption on the (^^-complementation ofF(n)(A) is false, and the proof is comp.ete. H

Defini t ion 2 .13 . An algebra is said to be a /Z2 — graded aigebra or a super-algebra ifA has two distinguished subspaces Ao and At satisfying the following conditions:

A - AQ ffi A, ; A\ , A\ C Ao ; A0Ay , A,A0 C .4, , (30)

in which case [Ao, Ai) is called a Z?2 — grading of A .

A locally convex space E is said to be distinguished, if for every bounded subset B\ inthe stong bidual E" of E , there exists a bounded subset B in E such that B\ C B"°,where B"" is the polar in E" of B" C E*. Hence any Banach space is distinguished.

P r o p o s i t i o n 2 . 1 ^ . Under the hypotheses of Theorem 2.9 , there are two closed idealsJ and R in F,n)(A) such that

F(n)(X) = J®R ; {0}=JR=R*. (31)

More precisely, J = [ei,e2 £„] , and R is the radical of F(n)(A") , whicli thereforesatisfies the Wedderburn Property.

Proof. From the definition of the ets and the F,s . it is clear that X satisfies (as aBanach space), the following direct sum:

= [eue2,---,tn\ ife, A'er(F0. (32)

Let J = [eue2, • • - ,£„) , and R = f |^ i A'er(F,) . Then J is a closed ideal since wehave

. FM(X) = J .

8

(33)

Moreover, according to Theorem 2.9 , each F, is a character, the kernel of which is aclosed ideal. Hence R is a closed ideal,

For u 6 J ; t>, w € R , it comes:

n

uv = y F,(u) FM ei = 0, (F;(u) = 0 V 1 < « < n) (34)

v w = £ ) F,<i>) F,(u>) e, = 0 , (F,(v) = F,(u>) = 0 , V 1 < i < n) (35)i=i

Since F(n)(X) is commutative, its (Jacobson) radical coincides with the intersection of thekernels of all its characters. Hence,

(36)Rad(FM(X)) = nr=i *'er(F.) = R ,

and the proof is complete. D

Corollary 2.15. F(n((Jf) is a 7Z2 - graded Banach algebra. D

From Proposition 2.14, we know what is the intersection of the kernels of all the irre-ducible representations of F(n)(X) . In what follows, we give the explicit expressions ofthe irreducible representations of F^(X)

Theorem 2.16. Let TT be an irreducible representation of F^(X) on some Banachspace M . Then t is given by

7r(i) = F(x) IM , (37)

where IM is the identity operator on M , and for some F £ {Fi, • •• ,F n}.

Proof. Let x be an irreducible representation of F(n)(A") on a Banach space M . ThenM admits a left F(n)(X)-module structure defined by

i m = *•(*) m ; x € F{n){X) , m € M. (38)

Let t € {1, • • • ,n} be given. Then for each m 6 M , we have

e.) m C ff(e.m). (39)

Hence for eacht € (1,2, •••,»»}, Mf(m) = (F(e,m) is a F(n)(^)-submodule of M , andaccording to the irreducibility of JT , we must have

Mi{m) = {0} , or Mi(m) = M. (40)

Assume first, that We, ma = M, for some m0 € M, and let w 6 M\{0}. Then thereexists A^gfZ" such that m = A^e.mo), and

w = x w = Xw(x e;) TOO =

It follows that

(41)

(42)

and therefore,ir(i) = F,(^)

where I\s denotes the identity operator on M .(43)

Now, suppose that (Fe(m = {0} for all m £ M, We choose in this case somej 6 {1,2, ••• ,n} for which there exists mi € M such that (Tfijtvi! ^ {0} , and thereforesatisfies: Ge^mi = M, according to the irreducibility of IT =ind since (Tejm is asubmodule of M . This implies according to what precedes, that w(x) — F}(X)IM, V I £X. But if such j does not exists, then

< T e , m = { 0 } V i € { l , 2 , - - - , n } and V i n e * / , (44)

In such a case, we get: J M = {0}. On the other hand, each a £ R and each m € Msatisfy

F[n)(X)(<Tam) = W(F(n)(X) a) m = {0} , (45)

that is,FM(X){R M) = {0} . (46)

R M is therefore a submodule of M and as such, is either equal to {0}, or coincides withM. But if R M = M, then

Fin){X) M = (J ® R) R M = {0} , (47)

and 7T would no longer be irreducible. So we necessarily must have R M = {0}, andagain

F[n){X) M = (J@R)(RM) = {0}, (48)

would contradict the choice of IT .

So we conclude that given i £ {1,2, • • • , n} , there always exist some tn0 € M , suchthat (Te mD = M , which gives consequently:

*{.) = F,( . ) / „ ,

and the proof is complete. •

Remark 2.17. In the decomposition Fideal J satisfies: J = F^jfJV) e , where

(49)

, in Proposition 2.14 above, the

-£«• (50)

is a unit for J and a modular unit for R . But if X is of dimension strictly greater thann , then F(nj(X) does not possess a unit nor a left (right) approximate identity.

Indeed, let (U , ) ,E / be a left approximate identity for F(n](X) . Then for all x € F[n)(X),we have:

10

_ ; . ! ! . :•-•

and it would follows that

contradicting the assumption on the dimension of X .Moreover, each e; is a minimal idempotent inducing a minimal idea! Jm, =(Te,- .

It is also clear that when dim(X) > n , then F^(X) is not semisimple, but itsradical fulfils: Rad{F(n){X))7 = {0}. We have the following:

Proposi t ion 2.18. Let A be a Banach algebra with a left approximate identity (tio)oEi,and radical R such that

A = J®R ; R2 = {0} . (53)

Then A is semisimple if and only if J is a closed two-sided ideal.

Proof. Condition (49) is clearly necessary. It is also sufficient as follows:

•42 = J 2 + J R +RJ. (54)

But J being a closed two-sided ideal, it comes

R J C J n R = {0} , and J R C J D R = {0} . (55)

Therefore A' = J2 C J • Moreover, since A admits a left approximate identity, thefollowing relations hold, where "clos( . )" stands for the norm closure.

A = closiA1) ; J = clos{J2) . (56)

Hence,A = clos(A2) = clos^) = J ,

which according to (53) , ensures the semisimplicity of A .

(57)

n

R e m a r k 2.19. The fact that F{n)(X) (n > 1) does not have an approximate identitycan be derived from Proposition 2.14 , for this assumption would make F(n)(.Y) fulfil allthe hypotheses of Proposition 2.16 , but as we know, F^(X) is not semisimple whendim(X) > 1.

The following result establishes a link between Fin)(X) (n > 1) and a general<F" -com piemen ted Banach algebra:

Theorem 2.20. Let A be a (H-complemented Banach algebra (n > 1), and call Bn,a subalgebra of A isomorphic to <Z"\ Then A can be given another Banach algebrastructure say F(n)(-4) , for which FM(A) is compact, commutative, and such that theproduct in F(n)(A) coincides on Bn , with the original product of A .

Proof. We may, without loss of generality, assume that Bn = [ei , -- ,en] , where(ei,Fi)1<i<n is an idempotent biorthogonal system associated to A . Then according toTheorem 2.9 , we endow A with a new product by setting:

Fi(b) a ; a , b € A. (58)

11

This product, together wii.h an adjustment of the norm in A , gives a Banach algebraF(o)(/1) which is commutative and compact. Moreover, for all a, b € B7l , we have

b = ^ F'(b) e, ,

and therefore,

a b = Fi(a) F,(b) £,- (prodv i in FM(A)) .

(59)

(00)

But since the elements e, are pairwise orthogonal idempotents of A , thi: product of aand b in A gives:

(61)

and according to relation (60) , coincides with their product in /')„)(/4) on Bn •DefinitlOD 2.21. If a Banach algebra is W"- complemented for some n > 1, we denote bym(A), the greatest integer such that A is (/""''''-complemented but not (^-complementedfor any k > m{A) , and call it the maximal complementation dimension of A.

If {ei, • • • ,em(A)} is a set of pairwise orthogonal idempotents, the Banach subalgebragenrerated by the e,s will be denoted by Bm(x) •

Definition 2.22. Let A be a Banach algebra with maximal complementation dimensionm(A) , and (e;, /w)i<i<m(jt) an idempotent biorthogonal system. Let e, € {0,1} for alliG {l ,---,m(/t)}. We"define

For each (ei,£j, • • • ,£m(/i)) € Pn[A), we let S° be the set uf all i such that £,= 0 , anddefine the subspace B(£),£2,- • • , £m(,4|) as follows:

F ( £ ( , £ i , • • • ,£m(, i)) = [e,]!€50 , (63)

where [ei];€so is the closed linear subspace generated by the e; , for i £ 5°, and we setby convention that:

B(l, •••,!) - {0} . (64)

Problem 2.23. According to Proposition 2.13 , the Wn- complemented Banach alge-bra F(n)(X) satisfies F(n)(jf) = [ei, • • •, en] © Rad (F(n)(X)). We know that an arbitrary(Fn-complemented complete normed algebra A satisfies A = [e;, • • • ,en] ® W, with someclosed subspace W . In what extent The radical of A may fulfil: Rad(A) C W 1

We only can prove the following:

Theorem 2.24. Let A be a Banach algebra and («,,F,)]<f<mW , Bm[A) , Pm{A] ,B(ei,t2, • • • ,£m(/i)) > be such as previously defined, and W^^ be the topological com-plement of 5m(4j . Then if Bm^ is an ideal, the radical of A satisfies the following

12

relation:

Rad(A)cA\ MJ'm(A)

where [},rw means the union over all two-sided ideals J contained in lVmiji.

Proof. Let A = Bm{A) © Wm(A) and J a two-sided ideal contained in Wmi.A)-(e*, /*i)i<;<m(,t> being an idempotent biorthogonal system, and Bm[A) being an ideal, the

(66)set

T = {Fu---,Fm{A)} ,

consists of multiplicative forms on the subalgebra

B = Bm(A] ® J . (67)

) and u , v € J , then x — xn -f u , and y = yn + vIndeed, if xn , yn €satisfy, for all F 6 T:

F(xy) = F(t , BB) = F[xn) F{yn)

A)since J C !VmM) C C\?J,

Now, assume that a € £ is quasi-invertibie with quasi-inverse b . Then b necessarilybelongs to E , and from

- a i ) = 0 , V

it follows thatF(a) ^ 1 , V FeT .

Moreover, it is checked without difficulty that each element

UMA)

• , £ - ( „ ) )

j € {1, • • • ,m(A)} ,

(68)

(69)

(70)

(71)satisfies

Fy(x)=l , /o

and is hence not quasi-invertible.

Therefore, (Jpm(jl) ( ^T=^ £ i c ' + ^ ' i ^ i i ' "" > em[A)) + ^) , does not meet the set q-Inv(A) of quasi-invertible elements of A , in virtue of (71) above. But according to [3];Proposition 18 , page 125 , Rad(A) is the union of all left (right) ideals all of whoseelements are quasi-invertible. The result then follows. D

Remark 2.25, It is worth noticing that for each two-sided ideal J C H ^ J J , the oper-ation of taking the complement in relation (65) giving the radical of A , does not affectany element of J , nor H^(j). More precisely, all the elements removed by the operation

13

of complementation fall outside H ^ j ^ .

Now, let A be a d"1 -complemented Banach algebra and let Bn and B'n ^ Bn besubalgebras of A , both isomorphic to ff". Then according to Theorem 2.18 , A admitstwo Banach algebra structures F(n)(A) and F'{n)(A) such that the products in F{n)(A)^ F(n)(

A) coincide on Bn ami B'n with the original product in A . A natural questionis: are F(n;(,4) and F^ isomorphic or not.

If 0 i s a n i s o m o r p h i s m f r o m F{n)(A) onto F{n){A), a n d if F[n)(A) = ( A ; Flt---, Fn) ,

F^(A) = (A; Gi, • • •, G^), then easy computations give

Fi = G, o Q ,

so that

We have

(72)

(73)

L e m m a 2.26. Let A be a (T2"-complemented Banach algebra. Then A admits threeBanach algebra structures say Au <42, and A3, satisfying:

Rad(A3) = Rad{Ai) and ~ A2 . (74)

Proof. Let (e,-, Fi)]<i<2n be a biorthogonal system. Then according to Proposition 2.13there exist Banach algebras A, = (A; Fu • • • ,Fn) , A3 = (A; Fn+1, • • -, F2n) , andA3 = (A;FU- • • ,F2n) satisfying

Rad{A3) = n^.A^rfF.)

n Rad{A2) .

On the other hand, A admits the topologicaJ direct sum

Hence each a 6 A fulfils:

€ fl?=i Keri.Fi) .

(75)

(76)

Define next a map ip from Aj to Ai as follows:

f : Ai i—y A

(77)

14

where a has the decomposition in (72).

[ ip is nothing else than the permutation, for a given i , of the coefficients ofe, and en+ l in the decomposition of a ],

For given x , y in A , we shall write (x y)\ , ( i y)2 and (x y)3 to means thatthe product xy is computed respectively in Aj , A? ,and A3 . So let a ,b £ A havedecompositions given by relation (77) , with respective xa and xb in f|^, A'er(Fj). Itcomes:

n

(« 4), = £ K(

The computation of ip(a) ip{b) gives in A?:

j'=n+t

(79)

that is, v1 ' s a n algebra morphism from A\ into A? • Moreover, the definition of ip yields

)=a ,V o f A , (80)

so that if is one-to-one and onto Aj , and hence an isomorphism, which completes theproof. D

T h e o r e m 2.27. Let A be a (P-complemented Banach algebra, Bn ^ B'n two subal-gebras of A such that Bn ~ <T ~ B'n . Let {e l l - - - ,e n} and {«i,---, t in} be sets ofpairwise orthogonal idempotents respectively in Bn and in B'n , and let F< , G; € A*be such that (e,,Fi)i<i<» > («i-,Gi}i<t<n are two idempotent biorthogonal systems. ThenF(n)(A) = {A;Fu---,Fn) is isomorphic to G[n)(A) = (/4; d , • • •, Gn)i<,<n , providedthe following condition is realized

Bncf)Ker(Gi) , and ffn C HL (81)

or equivalently if the system

e ! , e j , • • - , e n , u i , « 2 , - - • , « „ ; t\,- • • , r n , G i , • • - , G n ) ( S i )

is biorthogonal idempotent.

Proof. If the condition is fulfited, then setting en+it = u t ; 1 < i < n, we get a setC = {ej,-- • ,en ,en+i,- • • ,62n} be a set of 2n linearly independant elements of A , andcontinuous linear forms H, on A such that

fa) = 6i3 ; 1 < 1 , j < In (83)

15

twirt

Hi\[eu---,en] = F, ; 1 < i < n

//,[[en+i,---,e2r,] = G, ; n + 1 < i < 2ti .

It follows according to Remark 2.11 that

(84)

(85)

FM(A) = {A;Hi, • • - , # „ ) ; GW(>1) = ( A; / / „ + „ • • • , / / , „ ) . (86)

Hence, all the hypotheses of Lemma 2.26 are satisfied arid the result follows. •

The next result roughly says that if A is (^-complemented, then it admits ti algebrastructures, the radicals of which are ordered by inclusion.

Proposi t ion 2.28. Let A be a <£"-complemented Banach algebra. Then A can beturned in n different ways into compact commutative algebras, say /t t , A2, • • • , An suchthat:

D Rad(A?) D - - O iiarf(An)

and /JaJ(/lt)/iJo(/(/ln.m) is linearly isomorphic to a m-diinensional subspace of A(fc + m < n).

P r o o f . Let (e , ,F , ) ]< ,<<„ be an idempoten t biorthogonal sys t em. Given k 6 { l , - - - , n } ,we let A t = F{k)(A) as defined in Theorem 2.9 . T h e n /U satisfies A \ = [e, , • • • , e t j , andthe following relat ions are immedia te :

/ I , C A2 ^- C An . (SI)

According to Proposition 2.12, we have Rad{Ak) = HILi ^^(Fi)^ which also makesclear the announced ordering:

D Rad(A2) D---D Rad(An) . (88)

But the in-dimensional subspace [e t+ [, • • •, e t+m] being contained in Rad(Ak) , we have

Rad{Ak) =k+m

,efc+m] © f] Ker(F.)1 = 1

,e t + m ] © Rad{Ak+n) ,

which therefore gives:

and the proof is complete.

(89)

D

3 Normed algebras (^"-Complemented for all n > 1

Let i^lN) be the Banach algebra (under coordinatewise addition and multiplication) ofall sequences of complex numbers a = {a;),>i which are bounded in sup-norm: (Halloo =

16

Sup |a;| < oo). We havei>i

Example 3 .1 . too(IN) is a commutative Banach algebra which is (Un-complemented forall n > 1. Indeed, for each n > 1, the set {e,, ••• ,e»] in 4o(^V), where for each j , e} has1 for j " 1 coordinate and 0 elsewhere, consists of n pairwise orthogonal idempotents, whichaccording to Proposition 2.4 ensures that t^(IN) is (I1"-complemented for all n > 1. Wehave

Theorem 3.2. Let A be a Banach algebra <?"-complemented for all n > 1, and witha basic sequence of pairwise orthogonal idempotents. Then A contains a commutativesemisimple Banach subalgebra with a countable approximate identity.

Proof. Let (va),>\ be a basic sequence of pairwise orthogonal idempotents, and callB ro the Banach subspace of A linearly generated by the v,s . If (-F,);>i is the ascf of(«i)i>i ((«i).>i i s a b a s i s of S=°)> t n e n for i , «/ e Boo and for each n € IN, we have:

and since

it comes:

lim lli - V Fi{i) u,-|| = 0 ; lim \\y - V F,(j/) u,|| = 0 , (90)1=1 1=1

lim Mir) ".II = 0 ,

which, since (ui)t>l is a basis of B^ , gives:

" •

(91)

(92)

Relation (93) above describes the product of any two elements x , y in B^ , which istherefore a subalgebra of A.Now, it is easy to see from relation (93) that Bx is commutative and that each F\ is acharacter, so that

oo

Rad(Bx) C p|A'er(F,) = {0} , (93)

proving the semisimplicity of Bx .Finally, the sequence u = (en)n>i such that en = £ L i u, clearly satisfies

lim | |i - £„ x\\ = 0 ; x 6 B^ , (94)n—KM

17

which means that u is an approximate identity for Bm. •

Of course, the above Theorem admits no converse, as a consequence of tin- following:

Proposi t ion 3.3. Let K be a connected locally compact space (e.g. the space IR ofthe real numbers), and C(K) be the Banach space of all complex continuous functionson K . Then C(K) is a commutative semisimple Banach algebra with a unique nonzeroidempotent.

Proof. Let / € C(K) be idempotent. Then /2(() = f(t) , for all t € K, that is:f(t) G {0,1} , V t 6 K. But f being continuous, this condition cannot \iv fulfiled unlessK is the disjoint union of closed subsets [Ka)a^i such that given a £ /, one has:/(() = 0 , Or f(i) = 1 ; V t € Ka. But this cannot occur since K is connected. So theonly possibility is:

/(() = 0 V t € K or fit) = 1 V t <z A', (95)

which gives as only nonzero idempotent, the unit function 1 defined by:

= 1 ; V t € K (96)

and the proof is complete.

According to [2] Proposition 17 , page 166, each nonzero left ideal of a semisimpleBanach algebra contains a minimal idempotent. Theorem 3.2 may thercf.nc be viewed asa converse of this result, i.e. : To a. " large number " of minimal idempotents correspondsa semisimple (commutative) Bana.ch algebra.

On the other hand, it is proved in [2] , Proposition 16 , page 165 that If A is semi-prime({f}} is the only two-sided ideal J suck that J2 = {0}j and I is a closed two-sided ideal notin Rad(A) , then I contains a minima! closed two-sided ideal K with (K D Rad(A)) — {0}.

•F(n)(X) is an example of non semi-prime Banach algebra possessing this property asProposition 2.14 clearly shows.

Remark 3.4. Theorem 2.9 provides a tool in the construction of Banach algebras whoseradicals have finite codimension. It might be interesting to know if it is also possible toget similarly Banach algebras whose radicals are themselves nonzero finite dimensionalideals. As a matter of fact, each non semisimple finite dimensional Banach algebra an-swers positively this question. We particularly mention the following

Example 3.5. If X is a m-dimensional normed linear space, then for each n < m ,has finite dimensional radical: Rad(F^(X}) — [en+i,en+2,- • • ,em].

In the infinite dimensional case, the answer is positive as far as some Banach spacewith an unconditional basis is available, as it will soon be proved.

Definition 3.6. A series Yl^i Vi m a Banach space X is said to be unconditionallyconvergent, if the series £]^i SM>] ' s convergent for each permutation a of the index

18

. , • . ( !«•«. •

elements.

A basis (e,),>i in X is termed unconditional, provided each seriesunconditionally convergent.

X is

Proposi t ion 3.7. If (et).>i is an unconditional basis of a topological vector spaceX, then for each x — YlHi ^<e< anf^ e a c^ bounded scalar sequence (ct,),>i , the seriesx = ^T™, a,A,e, is convergent in X (cf. [1]). •

Theorem 3.8. A Banach space X with a basis, admits a dense subspace which can beturned into a commutative semisimple normed algebra (^"-complemented for all n > 1.[f the basis is unconditional, then the algebra product extends to the whole space X ,to give a commutative semisimple Banach algebra'1' ^"-complemented for all n > 1, andwith a countable approximate identity.

Proof. Let (e,-),->i be a basis of X with ascf (F,);>i, and let us define

Aa,(X)= | J [ e i , e 3 , - - , e n ] , (97)

and1

; Vi > l . (98)

Then Aoo(X) is clearly a dense subspace of X , and for all x , y € AX(X), there arep,q G IN such that x = £JL, F,(x)e{ ; y = £?_, Fi(y)e> • For s u c n x an(^ y , we set:

xy = Gfa) a , (99)

where m = min{p,q} and Gi — or/'2 F,,

Using " min - max " arguments shows that the above operation defines a product onAX(X) under which Aa,(X) is a commutative linear algebra. For i > 1, the elementsu, = oti~1e, satisfy

i ) e, == Q, ' e ; =

and similar computations lead to

u; u, = 0 , if i-f j .

Hence {ult uj, • • •, un, • • •} is a set of pairwise orthogonal idempotents, and for all n > 1,

Tkis result has been obtained by El-Helaly, S. k Husain, T. in [6}

19

it comes :

= [uu • • • , « „ ] ~ <Tn ,

that is, Ax(X) is (Tn-complemented for all n > 1. Moreover, for x and y such as in (100),we have:

< 11*11 M-Hence A^iX) is a normed algebra.

Now, let us assume that the basis (e,);>i is unconditional. Then

x H* ||x|| '= Sup l\Y] Fi(x)ei\\ , (100)

is a norm on X equivalent to its initial norm (cf. [7] ; page 33), so that there exists aconstant c > 0, such that

< Sup \\J2F,(x)e,\\ < c\\x\\. (101)

Let x — ^™1 F,(x) e, and y = £ ] " , Fi{y) e, be two arbitrary elements of X . Then thesequence (ajFi(ji))<>i is bounded since by an immediate application of (98) , it satisfies:

< 2 c (102)

Hence according to Proposition 3.7, the series £™, cr, Fi(x)Fi{y)e, converges to somez € X. We define for such x and y in X :

xy = • e, = ^ G , ( x ) G , ( j , ) e , .1=1

(103)

Relation (104) above extends the product on AX(X) to the whole Banach space X , whichtherefore becomes a commutative normed algebra that we denote by Aa,(X) , and whichis by construction W-complemented for all n > 1.

Now, the continuous linear forms H, — a^Fi ; * > 1 , satisfy for all x , y £ X :

j = i

= oc\ Fi(x) F\(y) = (a,Ft)(x) (a,F,

that is, each Hi is multiplicative on A^X) , and it is clear that if x F AX(X) satisfiesHi{x) = 0 , V i > 1, then x = 0. Moreover, since AX(X) is commutative, its radicalsatisfies:

Rad(A^(X)) C fj Ker(H,) = {0} , (104)

20

which means that Aoc(X) is semisimple. On the other hand, for all x , y €have:

) , we

Hence, / ^ ( X ) is a commutative semisimple Banach algebra <T"-complemented for alln > 1 , and it is an easy matter to check that the elements

an = ^Jti, ; n > 1 , (105)

provide an approximate identity for AX(X), and the proof is complete. D

Proposi t ion 3.9. Under the hypotheses and notations of Theorem 3.8 , AX{X) admitsan involution, continuous at least on on each of its finite dimensional subalgebras and forwhich its positive part satisfies:

fii*j € X ; &

Proof. Let us call i[/ the set of complex numbers of modulus one, and for eachx = J2%\ FAX)£] <= -V , let us define x' by

(106)

(107)

where F,(*) denotes the complex conjugate of Fj(x). The basis being unconditional, theseries in (108) converges to some z 6 X, since it satisfies:

(108)

with a bounded sequence (AJ(I))J>I defined as follows:

A , ( * ) = e j ( * ) e i ' ' M , «2 = - l ; J = 1 , 2 , - - .

where " e " is the exponential sign, and where

if

with

(109)

8j(x) E [0,2ir] ; t»'^ = ^ ( x J / F ^ i ) £ RJ , if F3(x) ? 0 (110}

2]

The map x i—> x" such defined is an invoiution on /4M(X) and is continuous on anysubalgebra of finite dimension N as follows, for each x = ^ =] F}(x)ej\

where cw = YliLi ll^jlllkjll • ^ac>1 sucn

N N

= * E (£>W e"J<") ^ = * E A>M «. i |A>U)| 6 {0. 1}

admits the following factorization:

. (Il l)

Now, direct computations show thai each e} is self-adjoint and satisfies

(112)

that is, e} is a positive element. It follows that.

which completes the proof. Q

We are now able to state the result which answers the question raised in Remark 3.4 :

Theorem 3.10. Let X be a Banach space with an unconditional basis. Then X givesrise to:(i) A commutative semisimple Banach algebra A] whose radical has finite codimension;(ii) A complete infinite dimensional normed algebra Ai with finite dimensional radical,(iii) Moreover. A^iX) being the the algebra of Theon in 3.8 , the following relation holds.

= Rad{At) e Rad(A2) (114)

Proof, (i) : Let x, y £ ^^(X) . Then according to Theorem 3.8, their product is givenby

x y = Y,Gi{x) Gi(y) e> ; G, = a]'1 F,- , (115)

where (ei)i>i '6 a n unconditional basis of X , with ascf {Fi)i>\. So the following operationsare well-defined products on A^X);

Pi(*,v)=Y,G<WGiMei ; P*(X>D) = Y, Gi{x) Gi(y) e, . (116)

The resulting complete normed algebras Ax and Ai satisfy respectively:

A, = (X;Gi,••-,(?»,) onrf A2 (X;Gn o + 1 ,Gn o + 2 , • • •) . (117)

22

According to Theorem 2.9 , Ai can be renormed in such a way to give a Banach algebra, which according to Proposition 2.13 has radical:

(118)

(119)

Rad(Fin)(X)) = f)Ke1 = 1

(ii) : Similarly, the radical of A2 satisfies

Rad(A2) = p | A'er(Gi) = [e,, • • • , e j

which consequently has finite dimension n0.

(iii) : X as a Ranach space satisfies

X = 8

which according to what precedes yields

AX{X) = Rad(Ax) © Rad{A2)

and the proof is complete. d

(120)

(121)

Acknowledgments

The author would like to thank Professor Abdus Salam, the International AtomicEnergy Agency and UNESCO for hospitality at the International Centre for TheoreticalPhysics, Trieste. He would also like to thank the Swedish Agency for Research Coop-eration with Developing Countries, SAREC, for financial support during his visit at theICTP under the Associateship scheme. He is aSso grateful to Professors M.S. Narasimhanand A. Verjovsky for their precious advice.

23

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[2] Bonsai 1, F.F. & Duncan, J. Complete normed algebra*. Spniigtr (\'f!3).

[3] Desquith, E. Banach algebra structures on the lopological dual of a lianaeh space,Afrika Matcmatika Series 3 vol. 3 (109!) 99-112.

[4] Desquith, E. Structures d'algebres de Banach sur h dual lopologiqiiF dun ispare, dtBanach et applications, These, dc doctoral d'Etat (FAST) Univerxilc d'Abidjan (1992).

[5] Desquith, E. An Arena product giving rise, to a weakly amenable Banach algebra, Pub-lications du Centre Univr.rsitaire du Luxembourg Travnux Mathe'maliques vol. V (1993)

[6] El-Helaly, S. h Husain, T. Unconditionality of orthogonal bases in Ho-algebras;Comm. Math. XXVIi 237-251 (1988),

[7] Husain, T. Orthogonal Schaudi r bases, Pure at d applied Mathematics Monographsvol. 143 (1991).

[8] Kothe, G. Topological victor spaces / , Springer Vtrlag band 159 (1969).

[9] Singer, I. Bases in Banach spaces, Springer Verlag (1970).