Approximate identities in Banach function algebras

H. G. Dales, Lancaster

National University of Ireland, Maynooth, 19 June 2013

In honour of the retirement of Anthony G. O’Farrell

Work in progress with Ali Ulger,¨ Istanbul

1 Definitions

Let K be a locally . Then C0(K) is the space of all complex-valued, continuous functions on K that vanish at infinity. This is a commutative with respect to the uniform | · |K.

A function algebra on K is a subalgebra of C0(K) such that, for x, y ∈ K with x 6= y, there is f ∈ A with f(x) 6= f(y), and, for each x ∈ K, there is f ∈ A with f(x) 6= 0.

A on K is a function algebra A that is a Banach algebra for a norm k · k, so that kfgk ≤ kfk kgk for f, g ∈ A.

Necessarily, kfk ≥ |f|K for f ∈ A.

The algebra A is a if it is closed in C0(K). 2 Natural Banach function algebras

A Banach function algebra A on K is natural if every character on A has the form f 7→ f(x) = εx(f) for some x ∈ K. Equiv- alently, every maximal modular ideal has the form

Mx = {f ∈ A : f(x) = 0} for some x ∈ K.

Every commutative, semisimple Banach algebra is a Banach function algebra on its character space.

3 Approximate identities

Let (A, k · k) be a natural Banach function algebra on K.

An is a net (eα) in A such that kfeα − fk → 0 for each f ∈ A.

A pointwise approximate identity is a net (eα) in A such that eα → 1 pointwise on K.

Clearly, each approximate identity is a point- wise approximate identity.

These approximate identities (eα) are bounded if sup keαk < ∞ (and the sup is the bound); they are contractive if the bound is 1. Say BAI and BPAI, CAI and CPAI.

The algebra A is (pointwise) contractive if every Mx has a C(P)AI.

We are interested in the best bound of BAIs. 4 Factorization

Let A be an algebra. Then

A[2] = {ab : a, b ∈ A},A2 = lin A[2] .

Cohen’s factorization theorem Let A be a commutative Banach algebra. Then the fol- lowing are equivalent:

(a) A has a BAI of bound m;

(b) for each ε > 0, each a ∈ A can be written as a = bc, where kbk ≤ m and ka − ck < ε (and so A = A[2], and A factors). 2

[Much more is true.]

5 Examples

(I) Take K compact and A = C(K). Then every Mx has a CAI.

Are there any more (pointwise) contractive uniform algebras?

(II) Let G be a LCA group (e.g., G = Z or 1 G = R), and let A be the group algebra (L (G),?). The Fourier transform maps L1(G) onto the Fourier algebra A(Γ), where Γ is the dual group to G.

Now A(Γ) is a natural Banach function algebra on Γ. The algebra A(Γ) always has a CAI, and all the maximal modular ideals Mγ have a BAI, of bound 2 (very standard); further, ‘2’ is the best bound of a BAI in Mγ (less well-known; Derighetti).

6 More examples

Look at natural Banach function algebras on I = [0, 1].

(III) Take A = Lip α(I), where α > 0, so that (|f(x) − f(y)|) kfk = |f| + sup α I |x − y|α for f ∈ A. If f(0) = 0 and f(1/n) = 1, then α kfk ∼ n , so there are no BPAI in M0.

(IV) Look at BVC(I), the algebra of continu- ous functions of bounded variation on I. Here kfk = |f| + var(f) var I for f ∈ BVC(I). There is an obvious BAI in M0 of bound 2.

7 The Choquet boundary

Let A be a Banach function algebra on K.

A closed subset F of K is a peak set if there exists a function f ∈ A with f(x) = 1 (x ∈ F ) and |f(y)| < 1 (y ∈ K \ F ); in this case, f peaks on F ; a point x ∈ K is a peak point if {x} is a peak set, and a p -point if {x} is an intersection of peak sets. The set of p -points of A is Γ0(A), the Choquet boundary of A. Its closure is the Silovˇ boundary.

Theorem Let A be a Banach function algebra on K. Then Mx has a BAI ⇒ x ∈ Γ0(A). 2

Example Let A be the disc algebra on D. Then {f ∈ A : f(0) = 0} does not have a BAI. This is obvious anyway. 2

8 Uniform algebras

Theorem Let A be a uniform algebra on com- pact K. Then the following are equivalent:

(a) x ∈ Γ0(A);

(b) Mx has a BAI;

(c) εx is an extreme point of the state space;

(d) Mx has a CAI.

The implication (a) ⇒ (d): Take f that peaks n at x. Then (1K − f : n ∈ N) is a BAI of bound n 2 for Mx. One can modify the functions f into functions gn ∈ A such that (1K − gn : n ∈ N) is a CAI for Mx. 2

Definition A natural uniform algebra on a com- pact space K is a Cole algebra if Γ0(A) = K.

Important fact There are Cole algebras other than C(K). 9 First question

Question What is the relation between the existence of a CPAI and a CAI in Banach function algebras?

Remark Suppose that there is a bounded net (eα) in A such that feα → f weakly for each f ∈ A. Then A has a BAI, with the same bound.

Original example – Jones and Lahr, 1977 +• Let S = (Q , +) be the semigroup of strictly positive rational numbers. The semigroup algebra A = (` 1(S),? ) is a natural Banach function algebra on Sb, the collection of semi- characters on S. Then there is a (nd) in such that (δ ) is a CPAI for A. However, N 1/nd A does not have any approximate identity.

This example is not pointwise contractive. 2 10 First question - more examples

Example Let G be a LCA group that is not compact. Then the minimum bound of a BAI in a maximal ideal of A(Γ) is 2. We can show that the minimum bound of a BPAI in a maximal ideal is 1 √ (1 + 2) > 1 , 2 so L1(G) is not pointwise contractive. We do not know if this constant is best-possible. 2

Example Let G be a LCA group that is not discrete. Let

n 1 1 o A = f ∈ L (G): fb ∈ L (Γ) , with kfk = max{kfk , fb }. Then A is a nat- 1 1 ural Banach function algebra on Γ. It has a CPAI, but no BAI. 2

11 First question - more examples

1 Example Let B = (L (R),?). Then there is a singular measure µ0 on R such that µ0 ? µ0 ∈ B (Hewitt). Look at A = B ⊕ Cµ0 as a closed subalgebra of (M(R),?). Then A has CPAI, but no approximate identity at all because A2 = B, which is not dense in A. (Hence CPAI 6⇒ factorization.) 2

Example Let A be a Banach function algebra that is reflexive as a . For exam- + ple, take the set of α = (αn) on Z such that

 ∞ 1/2 X 2 2 kαk =  |αn| (1 + n)  < ∞ , n=0 with convolution product, giving a Banach func- tion algebra on D.

Suppose that Mx has a BPAI. Then A contains the characteristic function of K \{x}, and so x is isolated. Thus maximal ideals of our exam- ple do not have BPAIs. 2 12 Uniform algebras

So far we have not given a pointwise contrac- tive Banach function algebra that is not con- tractive.

Fact Let A be a natural uniform algebra on K. Suppose that A has a BPAI (respectively, CPAI) that is a sequence. Then A has a BAI (respectively, CAI).

Proof Suppose that (fn) is bounded and fn → 1 pointwise on K. Then fn → 1 weakly by the dominated convergence theorem. An earlier remark now shows that A has a BAI, with the same bound. 2

13 Feinstein’s example

Example Joel Feinstein has an amazing exam- ple of a natural uniform algebra A on a com- pact, metrizable space K such that there is a point x0 ∈ K with Γ0(K) = K \{x0}.

The construction is a modification of Cole’s original construction.

Take M = Mx0. Each finite F disjoint from x0 is a peak set, and so there exists fF ∈ M that peaks on F . The net {fF } is a CPAI for M. All the other Mx have a CAI, so A is pointwise contractive. But M does not have a BAI because x0 is not a peak point, and so A is not contractive. (However M factors.) 2

14 Second question

Question Let A be a pointwise contractive, natural Banach function algebra. Is A neces- sarily a uniform algebra?

Let A be a contractive, natural Banach func- tion algebra. Is A necessarily a Cole algebra?

15 A special case

Definition Let S be a non-empty set. A Ba- nach sequence algebra on S is a Banach function algebra A on S such that c00(S) ⊂ A.

Theorem Let A be a pointwise contractive, natural Banach sequence algebra on a set S. Then A = c0(S).

This follows from a classical theorem of Bade and Curtis:

Theorem Let A be a Banach function algebra on a compact K. Suppose that there is m > 0 such that for disjoint, closed F and G in S, there is f ∈ A with |f|F < 1/2, |1 − f|G < 1/2, and with kfk ≤ m. Then A = C(K). 2

16 The BSE norm

Definition Let A be a natural Banach function algebra on K. Then L(A) is the closed linear 0 span of {εx : x ∈ K} as a subset of A , and

kfkBSE = sup{|hf, λi| : λ ∈ L(A)[1]} (f ∈ A) .

The abbreviation ‘BSE’ stands for Bochner– Schoenberg–Eberlein because of their theorem that characterizes the Fourier–Stieltjes trans- forms of the bounded Borel measures on LCA groups. The BSE-norm was introduced by Takahasi and Hatori in 1990.

Clearly

|f|K ≤ kfkBSE ≤ kfk (f ∈ A) .

Definition Let A be a natural Banach function algebra on K. Then A has a BSE norm if there is a constant C > 0 such that

kfk ≤ C kfkBSE (f ∈ A) .

17 Examples of BSE norms

(1) Trivially every uniform algebra has a BSE- norm.

(2) Let G be a LCA group and A = L1(G).

Then k · k1 = k · kBSE for A.

(3) Take α > 0, and consider A = Lipα(I) and A = lipα(I). Then k · kα = k · kBSE for A.

(4) Suppose that A is an ideal in (A00, 2) and A has a BPAI. Then A has a BSE norm. This covers Banach function algebras which are reflexive as Banach spaces, and all Banach sequence algebras on S with c00(S) dense.

18 A Banach sequence algebra with BSE norm

Here is another example of Feinstein.

Example For α = (αk) ∈ C N, say α ∈ A if 1 n kαk = |α| + sup X k α − α < ∞ . N k+1 k n∈N n k=1 Then A is a natural Banach sequence algebra on N.

Each maximal modular ideal of A has a BPAI of bound 4. However A2 is a closed subspace of infinite codimension in A, and so A does not have any approximate identity.

Clearly c00(S) is not dense in A. However again k · kBSE = k · k. We note that the Banach algebra A is not separable. 2

Query Do all natural Banach sequence algebras have a BSE norm? 19 Embarrassing fact

So far we have not actually found a Banach function algebra that does not have a BSE norm.

20 Specific open questions

Query Let Γ be a locally compact group, not necessarily abelian. There is a Fourier algebra A(Γ). It is a natural, strongly regular Banach function algebra on Γ.

The following are equivalent: (a) Γ is amenable; (b) A(Γ) has a BPAI; (c) A(Γ) has a BAI; (d) A(Γ) has a CAI. In these cases, A(Γ) has a BSE norm.

If Γ is not amenable, does A(Γ) have a BSE norm?

Query Let K and L be locally compact spaces, and set V (K,L) = C0(K)⊗b C0(L), the Varopou- los algebra.

Pn Here kF kπ = inf j=1 fj gj taken over all Pn K L representations F = j=1 fj ⊗ gj. This is the projective tensor norm.

We know that c0⊗b c0 has a BSE norm. Does V (K,L) always have a BSE norm? 21 A theorem

Proposition Suppose that A is pointwise con- √ tractive. Then kfkBSE ≤ 4 2 |f|K (f ∈ A). 2

Theorem Let A be a natural Banach function algebra on K such that A has a BSE-norm.

Suppose that A is pointwise contractive. Then A is a uniform algebra.

Suppose that A is contractive. Then A is a Cole algebra.

Proof Combine some earlier results. 2

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