banach function algebras and bse norms h. g. dales, lancaster · banach function algebras a banach...

Banach function algebras and BSEnorms

H. G. Dales, Lancaster

Joint work with Ali Ulger, Istanbul

Graduate course during 23rd Banach algebra

conference, Oulu, Finland

July 2017

g.dales@lancaster.ac.uk

1

Some references

H. G. Dales and A. Ulger, Approximate identi-

ties in Banach function algebras, Studia Math-

ematica, 226 (2015), 155–187.

H. G. Dales and A. Ulger, Banach function

algebras and BSE norms, in preparation.

E. Kaniuth and A. Ulger, The Bochner–Schoen-

berg–Eberlein property for commutative Ba-

nach algebras, especially Fourier and Fourier–

Stieltjes algebras, Trans. American Math. Soc.,

362 (2010), 4331–4356.

S.-E. Takahasi and O. Hatori, Commutative

Banach algebras that satisfy a Bochner–Schoen-

berg–Eberlein-type theorem, Proc. American

Math. Soc., 37 (1992), 47–52.

2

Banach spaces

Let E be a normed space. The closed unit ballis E[1] = x ∈ E : ‖x‖ ≤ 1.

The dual space of E is E′. This is the spaceof continuous = bounded linear functionals onE, and its norm is given by

‖λ‖ = sup |〈x, λ〉| = |λ(x)| : x ∈ E[1],so that (E′, ‖ · ‖) is a Banach space.

The weak-∗ topology on E′ is σ(E′, E). Thus(E′[1], σ(E′, E)) is compact.

The bidual of E is E′′ = (E′)′, and we regardE as a closed subspace of E′′; the canonicalembedding is κE : E → E′′, where

〈κE(x), λ〉 = 〈x, λ〉 (x ∈ E, λ ∈ E′).

For a closed subspace F of E, the annihilatorof F is

F⊥ = λ ∈ E′ : λ | F = 0 .

3

Algebras

All algebras are linear and associative and taken

over the complex field, C. The identity of

a unital algebra A is eA; the unitisation of a

(non-unital) algebra A is A].

For S, T ⊂ A, set

S · T = ab : a ∈ S, b ∈ T, ST = lin S · T ;

set A[2] = A · A and A2 = lin A[2].

An ideal in A is a linear subspace I such that

AI ⊂ I and IA ⊂ I.

4

The radical

We set A• = A \ 0; an element a of A is

quasi-nilpotent if zeA − a is invertible in A]

for each z ∈ C•, and the set of quasi-nilpotent

elements is denoted by Q(A).

The (Jacobson) radical of an algebra A is de-

noted by radA; it is the intersection of the

maximal modular left ideals; it is an ideal in A.

The algebra A is semi-simple if radA = 0and radical if radA = A, so that A is radical

if and only if A = Q(A).

5

Characters on algebras

A character = multiplicative linear functional

on an algebra A is a linear functional ϕ : A→ Csuch that ϕ(ab) = ϕ(a)ϕ(b) (a, b ∈ A) and also

ϕ 6= 0.

The character space of an algebra, the col-

lection of characters on A, is denoted by ΦA.

The centre of A is

Z(A) = a ∈ A : ab = ba (b ∈ A);

A is commutative if Z(A) = A.

6

Banach algebras

An algebra A with a norm ‖ · ‖ is a Banachalgebra (BA) if (A, ‖ · ‖) is a Banach space and

‖ab‖ ≤ ‖a‖ ‖b‖ (a, b ∈ A).

When A is unital, we also require that ‖eA‖ = 1.

Standard non-commutative example: A = B(E),the algebra of all bounded linear operators on aBanach space E, with operator norm ‖ · ‖op. Here

(ST )(x) = S(Tx) (x ∈ E)

for S, T ∈ A.

Each character ϕ on a BA is continuous, with‖ϕ‖ ≤ 1, and so ΦA ⊂ A′[1]; ΦA is a locally

compact subspace of (A′, σ(A′, A)), and it iscompact when A is unital.

In a BA, each maximal (modular) ideal is closedand so rad A is closed. Further

Q(A) = a ∈ A : limn→∞ ‖a

n‖1/n = 0.

7

Continuous functions

Let K be a locally compact space. Then Cb(K)is the algebra of all bounded, continuous func-tions on K, with the pointwise operations;

C0(K) consists of the continuous functions thatvanish at infinity;

C00(K) consists of the continuous functionswith compact support.

We define

|f |K = sup |f(x)| : x ∈ K (f ∈ Cb(K)) ,

so that | · |K is the uniform norm on K and(Cb(K), | · |K) is a commutative, semisimpleBanach algebra;

C0(K) is a closed ideal in Cb(K);

C00(K) is an ideal in Cb(K).

The topology of pointwise convergence on Cb(K)is called τp.

8

Function algebras

A function algebra on K is a subalgebra A of

Cb(K) that separates strongly the points of K,

in the sense that, for each x, y ∈ K with x 6= y,

there exists f ∈ A with f(x) 6= f(y), and, for

each x ∈ K, there exists f ∈ A with f(x) 6= 0.

Banach function algebras

A Banach function algebra (= BFA) on K

is a function algebra A on K with a norm ‖ · ‖such that (A, ‖ · ‖) is a Banach algebra.

The BFA A is natural if all characters on A

have the form εx : f 7→ f(x) for some x ∈ K;

equivalently, all maximal modular ideals are of

the form

Mx = ker εx = f ∈ A : f(x) = 0 .

9

Gel’fand theory

Let A be a BA. Define a(ϕ) = ϕ(a) (ϕ ∈ ΦA).

Then a ∈ C0(ΦA), and the Gel’fand transform

G : a 7→ a, (A, ‖ · ‖)→ (C0(ΦA), | · |ΦA),

is a continuous linear operator that is an alge-

bra homomorphism.

In the case where A is a CBA = commutative

Banach algebra, ker G = rad A = Q(A), and

so G is injective iff A is semi-simple.

Thus natural BFAs correspond to semi-simple

CBAs on their character space.

In the case where A is a commutative C∗-algebra, Gel’fand theory shows that A is iso-

metrically and algebraically ∗-isomorphic

to C0(ΦA).

10

More on BFAs

Henceforth K will be a non-empty, locally com-

pact (Hausdorff) space, and usually A will be

a natural BFA on K.

The closure of A ∩ C00(K) in A is called A0.

The BFA A is Tauberian if A = A0.

The ideal Jx in A consists of the functions in

A ∩ C00(K) that are 0 on a neighbourhood of

x, so that Jx ⊂Mx; A is strongly regular if Jxis dense in Mx for each x ∈ X.

11

Locally compact groups

Let G be a locally compact group with left

Haar measure mG. Then the group algebra

is (L1(G), ? , ‖ · ‖1) and the measure algebra

is (M(G), ? , ‖ · ‖), so that L1(G) is a closed

ideal in M(G). Both are semi-simple Banach

algebras. As a Banach space, M(G) = C0(G)′,and the product µ ? ν of µ, ν ∈ M(G) is given

by:

〈f, µ ? ν〉 =∫G

∫Gf(st) dµ(s) dν(t) (f ∈ C0(G)) .

The product of f, g ∈ L1(G) is given by

(f ? g)(t) =∫Gf(s)g(s−1t) dmG(s) (t ∈ G).

There is always one character on L1(G), namely

f 7→∫G f dmG; its kernel is the augmentation

ideal L10(G).

12

Dual Banach algebras

A Banach algebra A is a dual Banach algebra

if there is a closed submodule F of A′ such that

F ′ ∼ A, and then F is the predual of A. In this

case, we can write

A′′ = A⊕ F⊥

as a Banach space.

Key example: M(G) is a dual Banach algebra,

with predual C0(G).

13

Locally compact abelian groups

Let G be a locally compact abelian (LCA) group.

A character on G is a group homomorphism

from G onto the circle group T. The set Γ = G

of all continuous characters on G is an abelian

group with respect to pointwise multiplication

given by:

(γ1 + γ2)(s) = γ1(s)γ2(s) (s ∈ G, γ1, γ2 ∈ Γ) .

The topology on Γ is that of uniform cov-

ergence on compact subsets of G; with this

topology, Γ is also a LCA group, called the

dual group to G.

It is standard that the dual group of a compact

group is discrete and that the dual group of a

discrete group is compact.

For example, Z = T, T = Z, and R = R.

14

Pontryagin duality theorem

For each s ∈ G, the map

γ 7→ γ(s), Γ→ T,

is a continuous character on Γ, and the famous

Pontryagin duality theorem asserts that each

continuous character on Γ has this form and

that the topology of uniform convergence on

compact subsets of Γ coincides with the origi-

nal topology on Γ, so that Γ = G.

HenceG = G.

15

Fourier transform

Let G be a LCA group. The Fourier trans-

form of f ∈ L1(G) is f = Ff , so that

(Ff)(γ) = f(γ) =∫Gf(s)〈−s, γ〉dmG(s) (γ ∈ Γ) ,

and

A(Γ) =f : f ∈ L1(G)

,

is a natural, Tauberian BFA on Γ.

The Fourier–Stieltjes transform of µ ∈M(G)

is µ = Fµ, so that

(Fµ)(γ) = µ(γ) =∫G〈−s, γ〉dµ(s) (γ ∈ Γ) ,

and

B(Γ) = µ : µ ∈M(G) ,

is a Banach function algebra on Γ.

Of course, F : (M(G), ? ) → (B(Γ), · ) is a lin-

ear contraction that is an algebra isomorphism.

16

The group C∗-algebra

Here Γ is a locally compact group.

Let π be a representation of (L1(Γ), ? ), so that

π : L1(Γ)→ B(Hπ)

is a contractive ∗-homomorphism for some Hilbert

space Hπ. For f ∈ L1(Γ), define

|||f ||| = sup ‖π(f)‖ : π is a representation of L1(Γ) ,

so that |||f ||| ≤ ‖f‖1. Then ||| · ||| is a norm on

L1(Γ) such that

|||f∗ ? f ||| = |||f |||2 (f ∈ L1(Γ)) ,

and the completion of (L1(Γ), ||| · |||) is a C∗-algebra, called C∗(Γ), the group C∗-algebra

of Γ.

17

Fourier and Fourier–Stieltjes algebras

For a function f on a group Γ, we set

f(s) = f(s−1) (s ∈ Γ) .

Let Γ be a locally compact group.

The Fourier algebra on Γ is

A(Γ) = f ? g : f, g ∈ L2(Γ) .

Let Γ be a locally compact group. A functionf : Γ → C is positive-definite if it is continu-ous and if, for each n ∈ N, t1, . . . , tn ∈ G, andα1, . . . , αn ∈ C, we have

n∑i,j=1

αiαjf(t−1i tj) ≥ 0 .

The space of positive-definite functions on Γis denoted by P (Γ).

The Fourier–Stieltjes algebra on Γ, calledB(Γ), is the linear span of the positive-definitefunctions.

18

Properties of A(Γ) and B(Γ)

First, in the case where Γ is abelian, these twoalgebras agree with those previously defined.

Their theory originates in the seminal work ofEymard of 60 years ago.

The norm on B(Γ) comes from identifying itwith the dual of C∗(Γ), the group C∗-algebraof Γ.

For details of all this, see Lecture 1 of JorgeGalindo.

Theorem Let Γ be a locally compact group.Then A(Γ) is a natural, strongly regular, self-adjoint BFA on Γ, and B(Γ) is a self-adjointBFA on Γ. Further, A(Γ) is the closed ideal inB(Γ) that is the closure of B(Γ)∩C00(Γ). 2

Usually, A(Γ) ( B(Γ).

19

Facts about A(Γ) and B(Γ)

These facts will not be used, and terms are

not defined.

Facts A(Γ) is complemented in B(Γ); A(Γ) is

weakly sequentially complete; the dual space

A(Γ)′ is V N(Γ), the group von Neumann alge-

bra of Γ; A(Γ) is an ideal in its bidual iff Γ is

discrete. 2

Facts B(Γ) is a dual BFA, with predual C∗(Γ);

A(Γ) is a dual BFA iff A(Γ) = B(Γ) iff Γ is

compact [iff B(Γ) has the Schur property]. 2

20

Banach sequence algebras

Let S be a non-empty set, usually N. We write

c0(S) and `∞(S) for the Banach spaces of null

and bounded functions on S, respectively; the

algebra of all functions on S of finite support

is c00(S).

A Banach sequence algebra (= BSA) on S

is a BFA A on S such that

c00(S) ⊂ A ⊂ `∞(S) .

Thus A is Tauberian iff c00(S) is dense in A.

For example, ` p = ` p(N) and A(Z) with point-

wise product are Tauberian BSAs.

21

Biduals of Banach algebras

Let A be a Banach algebra. Then there are two

products 2 and 3 on A′′, the first and second

Arens products, that extend the given prod-

uct on A. Roughly:

Take M,N ∈ A′′, say M = limα aα and

N = limβ bβ, where (aα) and (bβ) are nets in A

(weak-∗ limits). Then

M2N = limα

limβaαbβ, M3N = lim

βlimαaαbβ .

The basic theorem of Arens is that κA : A→ A′′

is an isometric algebra monomorphism of A

into both (A′′,2) and (A′′,3).

We shall usually write just A′′ for (A′′,2).

22

Arens regularity

A Banach algebra A is Arens regular = AR if2 and 3 coincide on A′′. A commutative Ba-nach algebra is AR iff (A′′,2) is commutative.

Fact A′′ is a dual Banach algebra iff A is AR. 2

Let A be a C∗-algebra. Then A is AR and(A′′,2) is also a C∗-algebra, called the en-veloping von Neumann algebra.

In particular, (C0(K)′′,2) is a commutative C∗-algebra, and so has the form C(K) for a com-pact space K, called the hyper-Stonean en-velope of K. For K = N, we have K = βN,the Stone–Cech compactification of N.

Advertisement: this is discussed at length –with several ‘constructions’ and characteriza-tions of K – in

H. G. Dales, F. K. Dashiell, Jr., A. T.-M. Lau,and D. Strauss, Banach spaces of continuousfunctions as dual spaces, Springer, 2016

23

Strong Arens irregularity

Let A be a Banach algebra. Then the left andright topological centres are

Z(`)t (A′′)=

M ∈ A′′ : M2N = M3N (N ∈ A′′)

and

Z(r)t (A′′)=

M ∈ A′′ : N2M = N3M (N ∈ A′′)

,

respectively. Thus the algebra A is Arens reg-ular if and only if

Z(`)t (A′′) = Z

(r)t (A′′) = A′′ ;

A is strongly Arens irregular = SAI ifZ

(`)t (A′′) = Z

(r)t (A′′) = A.

In the case where A is commutative,

Z(`)t (A′′) = Z

(r)t (A′′) = Z(A′′).

Example Each group algebra L1(G) is SAI(Lau and Losert). Indeed, each measure al-gebra M(G) is SAI (Neufang et al). 2

24

Ideals in biduals

Let A be an algebra. For a ∈ A, we define La

and Ra by

La(b) = ab , Ra(b) = ba (b ∈ A) .

They are multipliers in an appropriate sense.

Let A be a BFA. Then A is an ideal in its

bidual if A is a closed ideal in A′′. This hap-

pens iff each La and Ra is a weakly compact

operator.

Fact Let A be a Tauberian BSA. Then Lf is

compact for each f ∈ A, and so A is an ideal

in its bidual. 2

There are non-Tauberian BSAs on N that are

ideals in their biduals, and there is a BSA on

N that is AR, but not an ideal in its bidual.

25

Tensor products

Let E and F be Banach spaces. Then (E ⊗F, ‖ · ‖π)

is their projective tensor product. Each ele-

ment z of E ⊗F can be expressed in the form

z =∞∑i=1

xi ⊗ yi ,

where xi ∈ E, yi ∈ F and∑∞i=1 ‖xi‖ ‖yi‖ < ∞,

and then ‖z‖π is the infimum of these sums

over all such representations.

The basic property of E ⊗F is the following:

for Banach spaces E, F , and G and each bounded

bilinear operator S : E × F → G, there is a

unique bounded linear operator TS : E ⊗F → G

such that TS(x ⊗ y) = S(x, y) (x ∈ E, y ∈ F )

and such that ‖TS‖ = ‖S‖.

26

Duals of tensor products

We have (E ⊗F )′ ∼= B(E,F ′), where the iso-

metric isomorphism

T : λ 7→ Tλ , (E ⊗F )′ → B(E,F ′) ,

satisfies the condition that

〈y, Tλx〉 = 〈x⊗y, λ〉 (x ∈ E, y ∈ F, λ ∈ (E ⊗F )′) .

This duality prescribes a weak-∗ topology on

B(E,F ′).

We use the following result of Cabello Sanchez

and Garcia:

Theorem Suppose that E′′ has the bounded

approximation property (BAP). Then the nat-

ural embedding of E ⊗F into (E ⊗F )′′ extends

to an isomorphic embedding of E′′ ⊗F ′′ onto a

closed subspace of (E ⊗F )′′. 2

27

Tensor products of BFAs

Let A and B be algebras, and set A = A⊗B.

Then there is a unique product on A with res-

pect to which A is an algebra and such that

(a1 ⊗ b1)(a2 ⊗ b2) = a1a2 ⊗ b1b2for a1, a2 ∈ A and b1, b2 ∈ B.

Fact Let A and B be natural BFAs on K and

L, respectively, and suppose that A has the ap-

proximation property. Then A ⊗B is a natural

BFA on K × L. If A has BAP, then A′′ ⊗B′′ is

a closed subalgebra of (A ⊗B)′′. 2

General question Suppose that A and B are

BFAs that are AR. Is A ⊗B AR?

A criterion involving biregularity and many ex-

amples (both ways) were given by Ali Ulger,

TAMS, 1988. See later.

28

Uniform algebras

A BFA A is a uniform algebra if it is closedin (Cb(K), | · |K), and so the norm is equivalentto the uniform norm.

For example, C0(K) is a natural uniform alge-bra on K. A natural uniform algebra A on K

is trivial if A = C0(K).

The disc algebra consists of all f analytic onD = z ∈ C : |z| < 1 and continuous on D.

A point x in K is a strong boundary pointfor A if, for each neighbourhood U of x, thereexists f ∈ A such that f(x) = |f |X = 1 and|f(y)| < 1 (y ∈ K \ U).

For x, y ∈ ΦA, say x ∼ y if ‖εx − εy‖ < 2. Thisis an equivalence relation that divides ΦA intoequivalence classes, called Gleason parts. Astrong boundary point is a singleton part, butnot conversely.

29

Approximate identities

Let A be a CBA. A net (eα) in A is an approx-imate identity for A if

limαaeα = a (a ∈ A) ;

an approximate identity (eα) is bounded ifsup α ‖eα‖ <∞, and then sup α ‖eα‖ is the bound;an approximate identity is contractive if it hasbound 1.

We refer to a BAI and a CAI, respectively, inthese two cases.

A natural BFA A on K is contractive if Mx

has a CAI for EACH x ∈ K.

Obvious example Take A = C0(K). Then Ais contractive. Are there any more contractiveBFAs? See later.

Group algebras have a CAI, but the augmen-tation ideal L1

0(G) has a BAI (of bound 2 - seelater), not a CAI, and so L1(G) is not contrac-tive.

30

Pointwise approximate identities

We shall consider (natural) BFAs on a locallycompact space K.

Let A be a natural BFA on K. A net (eα) in A

is a pointwise approximate identity (PAI) if

limαeα(x) = 1 (x ∈ K) ;

the PAI is bounded, with bound m > 0, ifsup α ‖eα‖ ≤ m, and then (eα) is a boundedpointwise approximate identity (BPAI); abounded pointwise approximate identity of bound1 is a contractive pointwise approximateidentity (CPAI).

Clearly a BAI is a BPAI and a CAI is a CPAI.

The algebra A is pointwise contractive if Mx

has a CPAI for each x ∈ K.

Also clearly a contractive BFA is pointwise con-tractive. But we shall give examples to showthat the converse is not true.

31

Contractive uniform algebras

Theorem Let A be a uniform algebra on a

compact space K, and take x ∈ K. Then the

following conditions on x are equivalent:

(a) εx ∈ exKA, where

KA = λ ∈ A′ : ‖λ‖ = 〈1K, λ〉 = 1 ;

(b) x is a strong boundary point;

(c) Mx has a BAI;

(d) Mx has a CAI.

Proof Most of this is standard.

(c) ⇒ (d) M ′′x is a maximal ideal in A′′, a closed

subalgebra of C(K)′′ = C(K). A BAI in Mx

gives an identity in M ′′x , hence an idempotent

in C(K). The latter have norm 1. So there is

a CAI in Mx. 2

32

Cole algebras

Definition Let A be a natural uniform algebra

on a compact space K. Then A is a Cole al-

gebra if every point of K is a strong boundary

point.

Theorem A uniform algebra is contractive if

and only if it is a Cole algebra. 2

There are non-trivial Cole algebras (but they

took some time to find). One is R(X) for a

certain compact set X in C2.

Theorem A natural uniform algebra on X is

pointwise contractive if and only if each set

x is a singleton Gleason part. 2

Standard examples now give separable uniform

algebra that are pointwise contractive, but not

contractive.33

The BSE norm

Definition Let A be a natural Banach functionalgebra on a locally compact space K. ThenL(A) is the linear span of εx : x ∈ K as asubset of A′, and

‖f‖BSE = sup |〈f, λ〉| : λ ∈ L(A)[1] (f ∈ A) .

Clearly K ⊂ L(A)[1] ⊂ A′[1], and so

|f |K ≤ ‖f‖BSE ≤ ‖f‖ (f ∈ A) .

In fact, ‖ · ‖BSE is an algebra norm on A - seelater.

Definition A BFA A has a BSE norm if thereis a constant C > 0 such that

‖f‖ ≤ C ‖f‖BSE (f ∈ A) .

Clearly each uniform algebra has a BSE norm.

A closed subalgebra of a BFA with BSE normalso has a BSE norm.

34

BSE algebras

Let A be a natural BFA on locally compact K.Then

M(A) = f ∈ Cb(K) : fA ⊂ A ,the multiplier algebra of A. It is a unital BFAon K with respect to the operator norm ‖ · ‖op.

For example, the multiplier algebra of L1(G) isM(G) (Wendel). This applies to all G: eachtwo-sided multiplier on (L1(G), ? ) has the formf 7→ f ? µ for some µ ∈M(G).

Let A be a natural Banach function algebra onK. Then

‖f‖BSE = sup |〈f, λ〉| : λ ∈ L(A)[1] (f ∈ Cb(K)) ,

and

CBSE(A) = f ∈ Cb(K) : ‖f‖BSE <∞ .The algebra A is a BSE algebra wheneverM(A) = CBSE(A). (It does not necessarilyhave a BSE norm.) For unital algebras, thecondition is that A = CBSE(A).

35

Basic theorem on CBSE(A)

The following is in TH in 1992.

Theorem Let A be a natural Banach func-tion algebra on K. Then (CBSE(A), ‖ · ‖BSE)is a Banach function algebra on K. Further,CBSE(A) is the set of functions f ∈ C b(K) forwhich there is a bounded net (fν) in A withlimν fν = f in (C b(K), τp); for f ∈ CBSE(A),the infimum of the bounds of such nets is equalto ‖f‖BSE.

Proof Certainly CBSE(A) is a linear subspaceof C b(K), and ‖ · ‖BSE is a norm on CBSE(A).It is a little exercise to check that(CBSE(A), ‖ · ‖BSE) is a Banach space.

Now take f1, f2 ∈ CBSE(A). We show that

‖f1f2‖BSE ≤ ‖f1‖BSE ‖f2‖BSE ;

we shall suppose that ‖f1‖BSE , ‖f2‖BSE = 1.

36

Proof continued

Take λ =∑ni=1αiεxi ∈ L(A)[1], and fix ε > 0.

First, set µ1 =∑ni=1αif1(xi)εxi, so that µ1 ∈ A′.

Then there exists g1 ∈ A[1] with |〈g1, µ1〉| >‖µ1‖ − ε. Next, set µ2 =

∑ni=1αig1(xi)εxi, so

that µ2 ∈ A′. Then there exists g2 ∈ A[1] with

|〈g2, µ2〉| > ‖µ2‖ − ε. We see that

〈f1, µ2〉 = 〈g1, µ1〉 and 〈g2, µ2〉 = 〈g1g2, λ〉 ,

and hence that |〈g2, µ2〉| ≤ ‖g1g2‖ ‖λ‖ ≤ 1. We

now have

|〈f1f2, λ〉| =

∣∣∣∣∣∣n∑i=1

αif1(xi)f2(xi)

∣∣∣∣∣∣= |〈f2, µ1〉| ≤ ‖µ1‖ < |〈g1, µ1〉|+ ε

= |〈f1, µ2〉|+ ε ≤ ‖µ2‖+ ε

< |〈g2, µ2〉|+ 2ε ≤ 1 + 2ε .

This holds for each λ ∈ L(A)[1] and each ε > 0,

and so ‖f1f2‖BSE ≤ 1, as required.

37

Proof concluded

Take f ∈ C b(K) to be such that there is a

bounded net (fν) in A[m] for some m > 0

such that limν fν = f in (C b(K), τp). For each

λ ∈ L(A)[1], we have

|〈f, λ〉| = limν|〈fν, λ〉| ≤ m,

and hence f ∈ CBSE(A)[m].

Conversely, suppose that f ∈ CBSE(A)[m], where

m > 0. For each non-empty, finite subset F

of K and each ε > 0, it follows from Helly’s

theorem that there exists fF,ε ∈ A such that

fF,ε(x) = f(x) (x ∈ F ) and∥∥∥fF,ε∥∥∥ ≤ m + ε.

Then the net (fF,ε) converges to f in

(C b(K), τp). 2

38

Sample general theorems – 1

Theorem Let A be a natural BFA. Then A is

a BSE algebra if and only if A has a BPAI and

the set

f ∈M(A) : ‖f‖BSE ≤ 1

is closed in (C b(ΦA), τp). 2

Theorem Let A be a natural BSA. Then CBSE(A)

is isometrically isomorphic to the Banach al-

gebra A′′/L(A)⊥, and CBSE(A) is a dual BFA,

with predual L(A). 2

Theorem Let A be a natural BFA. Then A

has a BSE norm iff the subalgebra A+ L(A)⊥

is closed in A′′. 2

39

Sample general theorems – 2

Theorem Let A be a dual BFA with predual

F . Suppose that the space ΦA∩F[1] is dense in

ΦA. Then A = CBSE(A) and ‖f‖ = ‖f‖BSE (f ∈ A).

Then A has a BSE norm.

Proof Take f ∈ CBSE(A), with ‖f‖BSE = m,

say. Then there is a bounded net (fν) in A[m]with limν fν = f in (C b(K), τp). Let g be an

accumulation point of this net in (A, σ(A,F )).

Then g(ϕ) = f(ϕ) (ϕ ∈ ΦA ∩ F[1]), and so

g = f . Thus f ∈ A[m] with ‖f‖ = ‖f‖BSE,

showing that A = CBSE(A). 2

Corollary Let G be a compact group. Then

M(G) has a BSE norm. 2

Theorem A BSE algebra has a BSE norm iff

it is closed in (M(A), ‖ · ‖op). 2

40

Sample general theorems – 3

The `1-norm on L(A) is given by∥∥∥∥∥∥n∑i=1

αiεxi

∥∥∥∥∥∥1

=n∑i=1

|αi| .

Theorem Let A be a BFA on K. Then

CBSE(A) = Cb(K) iff the usual norm on L(A)

is equivalent to the `1-norm. 2

41

Ideals in biduals

Theorem Let (A, ‖ · ‖) be a natural BFA on K

such that A is an ideal in its bidual. Then A isan ideal in CBSE(A) and

| · |K ≤ ‖ · ‖op ≤ ‖ · ‖BSE ≤ ‖ · ‖

on A. 2

Theorem (KU) Let A be a BFA that is an idealin its bidual. Then the following are equivalent:

(a) A is a BSE algebra;

(b) A has a BPAI;

(c) A has a BAI. 2

Theorem Let A be a dual BFA that is an idealin its bidual. Then A = CBSE(A) is AR, A hasa BSE norm, and A′′ = A⊕ L(A)⊥. 2

Theorem (*) Let A be a BFA that is an idealin its bidual, is AR, and has a BAI. Then A′′ isa BFA and has BSE norm. 2

42

Easy examples of BSAs

Here all algebras have coordinatewise products.

Example 1 Look at c0. Here c′′o = `∞, so c0 isan ideal in its bidual and is AR; it is not a dualalgebra. It has a BSE norm, and it is a BSEalgebra becauseM(c0) = `∞ = CBSE(c0). 2

Example 2 Look at `1, a Tauberian BSA, sothat `1 is an ideal in its bidual; it is a dual BSAwith predual c0. Here (`1)′ = `∞ = C(βN) and(`1)′′ = M(βN). Further, `1 = CBSE(`1) isAR, and M(βN) = `1nM(N∗), with the product

(α, µ)2 (β, ν) = (αβ,0) (α, β ∈ `1, µ, ν ∈M(N∗)) .

No BPAI, so not a BSE algebra; since L(`1)[1]is weak-∗ dense in (`1)′[1], the BSA `1 has aBSE norm. 2

Example 3 Look at ` p, where 1 < p <∞. Thisis a Tauberian BSA and is a reflexive Banachspace, and so ` p is an ideal in its bidual and adual algebra. It has a BSE norm, but it is nota BSE algebra. 2

43

General results

BSE algebras and BSE norms were introducedin 1990 by Takahasi and Hatori (TH) as anabstraction of a classical theorem of harmonicanalysis, the Bochner–Schoenberg–Eberleintheorem; see later.

Quite a few papers have discussed specific ex-amples. Our work seeks to give an underlyinggeneral theory, and applications to more exam-ples.

General questions Does every dual BFA havea BSE norm? Does every (even Tauberian)BSA have a BSE norm?

In both cases, we can give positive answerswith the help of modest extra hypotheses; wehave no counter-examples.

We can resolve these questions for many, butnot all, specific examples that we have lookedat – see below.

44

Contractive results

Theorem A contractive BFA with a BSE norm

is a Cole algebra. 2

Theorem Let A be a pointwise contractive

BFA with a BSE norm. Then the norms | · |Kand ‖ · ‖BSE on A are equivalent, and A is a

uniform algebra for which each singleton in ΦA

is a one-point Gleason part. Further, A is a

BSE algebra if and only if A = C(K). 2

Thus, to find (pointwise) contractive BFAs that

are not equivalent to uniform algebras, we must

look for those that do not have a BSE norm;

see later.

45

Queries for uniform algebras

Caution It is not true that every natural uni-

form algebra on a compact K is a BSE alge-

bra - a Cole algebra on a compact K is a BSE

algebra iff it is C(K), and so we can take a

non-trivial Cole algebra as a counter-example.

The disc algebra is a BSE algebra.

Query What is CBSE(A) for a uniform alge-

bra A? How do we characterize the uniform

algebras that are BSE algebras?

Query For example, what is CBSE(R(K)) for

compact K ⊂ C? Look at a Swiss cheese K.

46

Banach sequence algebras, bis

BSAs are more complicated than you mightsuspect. Does each natural BSA on N have aBSE norm? We have a general theorem thatat least covers the following example.

Example For α = (αk) ∈ CN, set

pn(α) =1

n

n∑k=1

k∣∣∣αk+1 − αk

∣∣∣ ,p(α) = sup pn(α) : n ∈ N .

Define A to be α ∈ c0 : p(α) <∞, so that Ais a self-adjoint BSA on N for the norm

‖α‖ = |α|N + p(α) (α ∈ A) .

Then A is a natural; A2 = A20 = A0 ( A; A is

not Tauberian; A is non-separable; A is not anideal in its bidual. The algebra A is not Arensregular.

This example does have a BSE norm, and it isa BSE algebra. 2

47

Tensor products of BSAs – 1

Guess Suppose that A and B are BFAs thatare BSE algebras/have BSE norms. Then A ⊗Bhas the corresponding property.

Let A and B be natural BSAs on S and T andsuppose that A has AP as a Banach space.Then A ⊗B is a natural BSA on S × T .

Example 1 Take p and q with 1 < p, q < ∞,and set A = ` p ⊗ ` q, so that A′ = B(` p, ` q

′).

Then A is a Tauberian BSA on N × N, and soan ideal in its bidual; it is the dual of K(` p, ` q

′);

it is AR.

It is reflexive iff pq > p+ q (Pitt) (this fails forp = q = 2).

Here A = CBSE(A), so A has a BSE norm, butA is not a BSE algebra. 2

48

Tensor products of BSAs – 2

Example 2 Let A = c0 ⊗ c0, so A is a Taube-

rian BSA on N×N, hence an ideal in its bidual.

It is AR, has a BSE norm, and it is a BSE

algebra. Here M(A) = CBSE(A) = A′′.

By Theorem (*), A′′ has a BSE norm.

Of course c′′0 = `∞ = C(βN); by an earlier

result, C(βN) ⊗C(βN) is a closed subalgebra

of A′′, and so also has a BSE norm.

We do not know if either C(βN) ⊗C(βN) or A′′

is a BSE algebra.

Neufang has shown that A′′ is not AR - see

his lecture. What about C(βN) ⊗C(βN)?

49

Varopoulos algebra

Let K and L be compact spaces, and set

V (K,L) = C(K) ⊗C(L) ,

the projective tensor product of C(K) and C(L);

this algebra is the Varopoulos algebra.

It is a natural, self-adjoint BFA on K×L, dense

in C(K ×L). The dual space is identified with

B(C(K),M(L)).

To show that V = V (K,L) has a BSE norm,

we must show that L(V )[1] is weak-∗ dense in

V ′[1] = B(C(K),M(L))[1]:

given T ∈ B(C(K),M(L))[1], ε > 0, n ∈ N,

f1, . . . , fn ∈ C(K), and g1, . . . , gn ∈ C(L), we

must find S ∈ L(V )[1] such that

|〈g, (T − S)f〉| < ε

whenever f ∈ f1, . . . , fn and g ∈ g1, . . . , gn.50

Varopoulos algebra, continued

We can do this by choosing suitable partitions

of unity in C(K) and C(L). Thus:

Theorem For compact K and L, V (K,L) has

a BSE norm. 2

Question Is V = V (K,L) a BSE algebra?

For this, we would have to show that CBSE(V ) = V .

At least we know that CBSE(V ) ( C(K × L),

using a result in the book of Helemskii.

51

Tensor products of uniform algebras

Let A and B be natural uniform algebras on

K and L, respectively. It is natural to ask if

A ⊗B always has a BSE norm. Clearly this

would follow immediately from the above if we

knew that A ⊗B were a closed subalgebra of

V (K,L). However this is not easily seen: it

is not immediate because a proper uniform al-

gebra A on a compact space K is never com-

plemented in C(K). The result is true in the

special case where A and B are the disc alge-

bra, as shown by Bourgain

Theorem Let A := A(D) to be the disc al-

gebra. Then A ⊗A is a closed subalgebra of

V (D,D), and so A ⊗A has a BSE norm. 2

Query What happens for different uniform al-

gebras? Is A(D) ⊗A(D) a BSE algebra?

52

Group and measure algebras

Let G be an infinite, LCA group with dual Γ.

Theorem (i) L1(G) is a BSE algebra, and

M(L1(G)) = CBSE(L1(G)) = M(G) .

(ii) ‖µ‖ = ‖µ‖BSE (µ ∈ M(G)), and so M(G)

and L1(G) each have a BSE norm.

(iii) M(G) is a BSE algebra iff G is discrete.

Proof (i) Classical Bochner–Schoenberg–Eberlein

theorem.

(ii) Uses almost periodic functions on G.

(iii) It is easy to find functions in CBSE(M(G))

that are not in M(G) when G is not discrete. 2

53

Compact abelian groups

Take G to be a compact, abelian group. For

1 ≤ p ≤ ∞, (Lp(G), ? ) is a semi-simple CBA.

For 1 < p <∞, F(Lp(G), ? ) is a Tauberian BSA

on Γ; it is reflexive; and hence AR and an ideal

in its bidual and a dual BFA; it does not have

a BPAI.

Further, F(L∞(G), ? ) is a natural BSA on Γ,

but it is not Tauberian. It is a dual BSA with

predual A(Γ); it is AR; it is an ideal in its bid-

ual.

Theorem For 1 < p ≤ ∞, F(Lp(G)) has a BSE

norm, but it is not a BSE algebra. 2

54

Beurling algebras on Z

A weight on Z is a function ω : Z → [1,∞)

such that ω(0) = 1 and

ω(m+ n) ≤ ω(m)ω(n) (m,n ∈ Z) .

Then `1(Z, ω) is the space of functions

f =∑f(n)δn such that

‖f‖ω =∑|f(n)|ω(n) <∞ .

This is a commutative Banach algebra for con-

volution. Via the Fourier transform, `1(Z, ω) is

a BFA on the circle or an annulus in C.

The algebra `1(Z, ω) is a dual BFA, with pre-

dual c0(Z,1/ω); it is not an ideal in its bidual.

Examples show that `1(Z, ω) may be AR, that

it may be that ω is unbounded and it is SAI; it

may be neither AR nor SAI (D-Lau).

55

Beurling algebras as BSE algebras

Theorem Beurling algebras Aω are BSE alge-

bras with a BSE norm for most, may be all,

weights.

Proof This works when ΦAω ∩ c0(Z,1/ω)[1] is

dense in ΦAω. 2

Trouble for weights ω with lim supn→∞ ω(n) =∞and lim infn→∞ ω(n) = 1; they exist.

56

Figa-Talamanca–Herz algebras

Let Γ be a locally compact group.and take p

with 1 < p < ∞. The Figa-Talamanca–Herz

(FTH) algebra is Ap(Γ). Formally, Ap(Γ) is

the collection of sums

f =∞∑n=1

gn ? hn

where gn ∈ Lp(Γ) and hn ∈ Lp′(Γ) for each

n ∈ N and∑∞n=1 ‖gn‖p ‖hn‖p′ < ∞, and ‖f‖ is

the infimum of such sums.

Thus Ap(Γ) is a self-adjoint, Tauberian, nat-

ural, strongly regular Banach function algebra

on Γ.

[See papers of Herz, a book and lectures of

Derighetti.]

57

BAIs and BPAIs in FTH algebras

Theorem (mainly Leptin) Let Γ be a locally

compact group, and take p > 1. Then the

following are equivalent:

(a) Γ is amenable;

(b) Ap(Γ) has a BAI;

(c) Ap(Γ) has a BPAI;

(d) Ap(Γ) has a CAI. 2

58

Arens regularity of Fourier algebras

Theorem (Lau–Wong) Let Γ be a LC group,

and suppose that A(Γ) is AR. Then Γ is dis-

crete, and every amenable subgroup is finite.

May be Γ must be finite. 2

Suppose that Γ is discrete. If Γ is amenable,

then A(Γ) is SAI (Lau–Losert, 1988), but not

if Γ contains F2 (Losert, 2016).

For the case where Γ is not discrete, and es-

pecially when Γ is compact, see the lectures of

Jorge Galindo in Oulu.

59

BSE properties

Theorem (essentially Eymard) Let Γ be a LC

group. Then

‖f‖ = ‖f‖BSE (f ∈ B(Γ)) ,

and so A(Γ) and B(Γ) have a BSE norm.

Proof Since B(Γ) = C∗(Γ)′, we can use Ka-

plansky’s density theorem for C∗-algebras. 2

Theorem (KU) A(Γ) is a BSE algebra iff Γ is

amenable. 2

60

B(Γ) as a BSE algebra

Let Γ be a LC group.

In the case where Γ is compact, A(Γ) = B(Γ),

and so B(Γ) is a BSE algebra.

In the case where Γ is not compact, there is,

as shown in KU, surprising diversity: there are

amenable groups for which B(Γ) is and is not

a BSE algebra, and there are non-amenable

groups for which B(Γ) is and is not a BSE

algebra.

61

Tensor products of Fourier algebras

Let Γ1 and Γ2 be locally compact groups. Sup-

pose that

A(Γ1) ⊗A(Γ2) = A(Γ1 × Γ2) . (∗)

Then A(Γ1) ⊗A(Γ2) has a BSE norm, and it

is a BSE algebra if and only if both Γ1 and

Γ2 are amenable. But (*) is not always true

(Losert).

Guess A(Γ1) ⊗A(Γ2) always has a BSE norm,

and is a BSE algebra if and only if both Γ1 and

Γ2 are amenable.

62

BAIs and BPAIs in maximal ideals ofFourier algebras

Let Γ be an infinite, amenable locally compactgroup, and let M be a maximal modular idealof A(Γ).

It is standard that M has a BAI of bound 2.By a theorem of Delaporte and Derighetti,the number 2 is the minimum bound for sucha BAI. We now consider pointwise versions ofthis.

Theorem Let Γ be an infinite locally compactgroup such that Γd is amenable. Then theminimum bound of a BPAI in M is also 2. Inparticular, A(Γ) is not pointwise contractive.2

Query What happens if Γ is amenable, butΓd is not? (Eg., Γ = SO(3).) The minimumbound is > 1.

Query What happens for Ap(Γ) when p > 1and p 6= 2?

63

FTH algebras Ap(Γ)

Here Γ is a LC group and 1 < p <∞.

Theorem (Forrest) Ap(Γ) is an ideal in its bi-

dual iff Γ is discrete. 2

Theorem (Forrest) Suppose that Ap(Γ) is AR.

Then Γ is discrete and every abelian subgroup

is finite. May be Γ must be finite. 2

Apparently nothing is known of when Ap(Γ) is

SAI.

There are varying definitions of Bp(Γ). The

first was by Herz. Cowling said it wasM(Ap(Γ));

Runde gave a definition involving representa-

tion theory; we prefer Runde’s definition be-

cause it gives the previous Bp(Γ) when p = 2.

The definitions all agree when Γ is amenable.

64

BSE properties of FTH algebras

This is harder than for the case p = 2 because

we have no help from C∗-algebra theory. Take

p with 1 < p <∞.

Theorem Let Γ be a locally compact group.

Then Ap(Γ) is a BSE algebra if and only if Γ

is amenable. In this case,

Bp(Γ) = CBSE(Ap(Γ)) =M(Ap(Γ)) ,

and Ap(Γ) and Bp(Γ) have BSE norms.

Proof Uses interplay with Bp(Γd) and results

of Herz and of Derighetti. 2

Query Does Ap(Γ) have a BSE norm for each

Γ? This is true for p = 2.

65

Segal algebras

Definition Let (A, ‖ · ‖A) be a natural Banachfunction algebra on a locally compact spaceK. A Banach function algebra (B, ‖ · ‖B) is anabstract Segal algebra (with respect to A) ifB is an ideal in A and there is a net in B thatis an approximate identity for both (A, ‖ · ‖A)and (B, ‖ · ‖B).

Classical Segal algebras are abstract Segalalgebras with respect to L1(G).

Let S be a Segal algebra with respect to L1(G).Then F(S) is a natural, Tauberian BFA on Γ;it is an ideal in its bidual iff G is compact.

The norm is equivalent to ‖ · ‖1 iff S = L1(G).

Always M(G) ⊂M(S) (but not necessarily equal).

Theorem A Segal algebra S is a BSE algebraiff S has BPAI, and then

M(G) =M(S) = CBSE(S) . 2

66

BSE norms for Segal algebras

Let S be a Segal algebra on a LC group G.

Suppose that S has a CPAI. Then we can iden-

tify the BSE norm.

Indeed, for f ∈ S, we have∣∣∣f ∣∣∣Γ≤ ‖f‖BSE,S = ‖f‖op,S = ‖f‖1 ≤ ‖f‖S ,

and so S has a BSE norm iff S = L1(G).

67

An example of a Segal algebra

Example Let G be a non-discrete LCA group

with dual group Γ. Take p ≥ 1, define

Sp(G) = f ∈ L1(G) : f ∈ Lp(Γ) ,

and set

‖f‖Sp = max‖f‖1 ,

∥∥∥f ∥∥∥p

(f ∈ Sp(G)) .

Then (Sp(G), ? , ‖ · ‖Sp) is a Segal algebra with

respect to L1(G) and a natural, Tauberian BFA

on Γ. Since Sp(G)2 ( Sp(G), Sp(G) does not

have a BAI. However, by a result of Inoue and

Takahari, Sp(R) has a CPAI.

Thus Sp(R) is a BSE algebra without a BSE

norm. 2

68

Final example

Example We give a BFA A on the circle T, butwe identify C(T) with a subalgebra of C[−1,1].We fix α with 1 < α < 2.

Take f ∈ C(T). For t ∈ [−1,1], the shift of fby t is defined by

(Stf)(s) = f(s− t) (s ∈ [−1,1]) .

Define

Ωf(t) = ‖f − Stf‖1 =∫ 1

−1|f(s)− f(s− t)| ds

and

I(f) =∫ 1

−1

Ωf(t)

|t|αdt .

Then A = f ∈ C(T) : I(f) <∞ and

‖f‖ = |f |T + I(f) (f ∈ A) .

We see that (A, ‖ · ‖) is a natural, unital BFAon T; it is homogeneous.

69

Final example continued

Let en be the trigonometric polynomial given

by en(s) = exp(iπns) (s ∈ [−1,1]). Then

en ∈ A, and so A is uniformly dense in C(T).

But ‖en‖ ∼ nα−1, and so (A, ‖ · ‖) is not equiv-

alent to a uniform algebra.

We claim that A is contractive. We show that

M := f ∈ A : f(0) = 0 has a CAI.

For this, define

∆n(s) = max 1− n |s| ,0 (s ∈ [−1,1], n ∈ N) .

Then we can see that I(∆n) ∼ 1/n2−α, and so

‖1−∆n‖ ≤ 1+O(1/n2−α) = 1+o(1). Further,

a calculation shows that (1−∆n : n ∈ N) is an

approximate identity for M .

We conclude that ((1−∆n)/ ‖1−∆n‖ : n ∈ N)

is a CAI in M , and so A is contractive. 2

70

Conclusions

We have a contractive BFA not equivalent

to a uniform algebra.

Here the BSE norm is equal to the uniform

norm, and CBSE(A) = C(I), whereas M(A) = A,

so A is not a BSE algebra.

Thus our example is neither a BSE algebra nor

has a BSE norm.

71