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 compact space. Then C0(K) is the space of all complex-valued, continuous functions on K that vanish at infinity. This is a commutative Banach algebra with respect to the uniform norm j · jK. A function algebra on K is a subalgebra of C0(K) such that, for x; y 2 K with x 6= y, there is f 2 A with f(x) 6= f(y), and, for each x 2 K, there is f 2 A with f(x) 6= 0. A Banach function algebra 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 2 A. Necessarily, kfk ≥ jfjK for f 2 A. The algebra A is a uniform algebra 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) = "x(f) for some x 2 K. Equiv- alently, every maximal modular ideal has the form Mx = ff 2 A : f(x) = 0g for some x 2 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 approximate identity is a net (eα) in A such that kfeα − fk ! 0 for each f 2 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 < 1 (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] = fab : a; b 2 Ag;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 " > 0, each a 2 A can be written as a = bc, where kbk ≤ m and ka − ck < " (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 (jf(x) − f(y)j) kfk = jfj + sup α I jx − yjα for f 2 A. If f(0) = 0 and f(1=n) = 1, then α kfk ∼ n , so there are no BPAI in M0. (IV) Look at BV C(I), the algebra of continu- ous functions of bounded variation on I. Here kfk = jfj + var(f) var I for f 2 BV C(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 2 A with f(x) = 1 (x 2 F ) and jf(y)j < 1 (y 2 K n F ); in this case, f peaks on F ; a point x 2 K is a peak point if fxg is a peak set, and a p -point if fxg 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 2 Γ0(A). 2 Example Let A be the disc algebra on D. Then ff 2 A : f(0) = 0g 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 2 Γ0(A); (b) Mx has a BAI; (c) "x 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 2 N) is a BAI of bound n 2 for Mx. One can modify the functions f into functions gn 2 A such that (1K − gn : n 2 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 2 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 sequence (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 p (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 2 L (G): fb 2 L (Γ) ; with kfk = maxfkfk ; fb g. 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 2 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 Banach space. For exam- + ple, take the set of sequences α = (αn) on Z such that 0 1 11=2 X 2 2 kαk = @ jαnj (1 + n) A < 1 ; 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 n fxg, 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 2 K with Γ0(K) = K n fx0g. 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 2 M that peaks on F . The net ffF g 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.