Amenability properties of Banach algebra valued continuous functions Fields workshop, Toronto, May 24, 2014 Yong Zhang based on joint work with R. Ghamarshoushtari Department of Mathematics University of Manitoba, Winnipeg, Canada Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 1 / 23 Outline 1 Preliminaries 2 Amenability 3 generalized amenability 4 weak amenability Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 2 / 23 C(X; A) Let X be a compact Hausdorff space and A a Banach algebra. Denote C(X:A) = the space of A-valued continuous functions on X: With pointwise algebraic operations and the uniform norm kf k1 = supfkf (x)kA : x 2 Xg C(X; A) is a Banach algebra. Examples 1 P C(X; `1) = f(xi (t)) : xi 2 C(X); jxi j converges uniformly on Xg i=1 Let M be a W*-algebra and E be its predual, Then C(X; M) = K(E; C(X)), the space of compact operators from E into C(X). Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 3 / 23 It is reasonable to expect that, normally, A rather than X plays the decisive role in the structure of C(X; A). We are concerned with the amenability properties of C(X; A). We will show constructively, among other things, that C(X; A) is amenable if and only if A is amenable; if A is commutative, then C(X; A) is weakly amenable if and only if A is weakly amenable. Early investigation of C(X; A) goes back to 1940’s, when I. Kaplansky and A. Hausner studied the maximal ideal space of the algebra for commutative A. We note C(X; A) is a C*-algebra if and only if A is a C*-algebra. C(X; A) is commutative if and only if A is commutative. C(X; A) has a BAI if and only if A has a BAI. Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 4 / 23 We are concerned with the amenability properties of C(X; A). We will show constructively, among other things, that C(X; A) is amenable if and only if A is amenable; if A is commutative, then C(X; A) is weakly amenable if and only if A is weakly amenable. Early investigation of C(X; A) goes back to 1940’s, when I. Kaplansky and A. Hausner studied the maximal ideal space of the algebra for commutative A. We note C(X; A) is a C*-algebra if and only if A is a C*-algebra. C(X; A) is commutative if and only if A is commutative. C(X; A) has a BAI if and only if A has a BAI. It is reasonable to expect that, normally, A rather than X plays the decisive role in the structure of C(X; A). Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 4 / 23 Early investigation of C(X; A) goes back to 1940’s, when I. Kaplansky and A. Hausner studied the maximal ideal space of the algebra for commutative A. We note C(X; A) is a C*-algebra if and only if A is a C*-algebra. C(X; A) is commutative if and only if A is commutative. C(X; A) has a BAI if and only if A has a BAI. It is reasonable to expect that, normally, A rather than X plays the decisive role in the structure of C(X; A). We are concerned with the amenability properties of C(X; A). We will show constructively, among other things, that C(X; A) is amenable if and only if A is amenable; if A is commutative, then C(X; A) is weakly amenable if and only if A is weakly amenable. Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 4 / 23 If A is a Banach algebra, then A⊗A^ is a Banach A-bimodule with the module actions determined by a · (b ⊗ c) = ab ⊗ c; (b ⊗ c) · a = b ⊗ ca Definition 1 A net (αν)) ⊂ A⊗A^ is called an approximate diagonal for A if lim ka · αν − αν · akp = 0 and lim π(αν)a = a (a 2 A); ν ν where π: A⊗A^ ! A is the product map defined by π(a ⊗ b) = ab. If in addition there is constant m > 0 such that kανk ≤ m for all ν, then (αν) is called a bounded approximate diagonal. approximate diagonal For Banach spaces V and W , we denote by V ⊗ W the algebraic tensor product, and by V ⊗^ W the Banach space projective tensor product of V and W . The norm of V ⊗^ W is denoted by k · kp. Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 5 / 23 approximate diagonal For Banach spaces V and W , we denote by V ⊗ W the algebraic tensor product, and by V ⊗^ W the Banach space projective tensor product of V and W . The norm of V ⊗^ W is denoted by k · kp. If A is a Banach algebra, then A⊗A^ is a Banach A-bimodule with the module actions determined by a · (b ⊗ c) = ab ⊗ c; (b ⊗ c) · a = b ⊗ ca Definition 1 A net (αν)) ⊂ A⊗A^ is called an approximate diagonal for A if lim ka · αν − αν · akp = 0 and lim π(αν)a = a (a 2 A); ν ν where π: A⊗A^ ! A is the product map defined by π(a ⊗ b) = ab. If in addition there is constant m > 0 such that kανk ≤ m for all ν, then (αν) is called a bounded approximate diagonal. Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 5 / 23 For a locally compact group G, B.E. Johnson (1972) showed that L1(G) is amenable if and only if G is an amenable group. Using Johnson’s above result on L1(G) and the Stone-Weierstrass Theorem, M. V, Sheinberg (1977) showed that C(X) = C(X; C) is amenable for any compact Hausdorff space X. A direct proof for the amenability of C(X), by constructing a bounded approximate diagonal, was give by (Abtahi-Z. 2010). We are concerned with general C(X; A). amenability A Banach algebra is called amenable if there is a bounded approximate diagonal for it. Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 6 / 23 amenability A Banach algebra is called amenable if there is a bounded approximate diagonal for it. For a locally compact group G, B.E. Johnson (1972) showed that L1(G) is amenable if and only if G is an amenable group. Using Johnson’s above result on L1(G) and the Stone-Weierstrass Theorem, M. V, Sheinberg (1977) showed that C(X) = C(X; C) is amenable for any compact Hausdorff space X. A direct proof for the amenability of C(X), by constructing a bounded approximate diagonal, was give by (Abtahi-Z. 2010). We are concerned with general C(X; A). Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 6 / 23 Grothendieck inequality The following inequality due to A. Grothendieck is important to us. Theorem 1 (Grothendieck) Let K1; K2 be compact Hausdorff spaces, and let Φ be a bounded scalar-valued bilinear form on C(K1) × C(K2). Then there are probability measures µ1; µ2 on K1; K2, respectively, and a constant k > 0 such that 1 Z Z 2 2 2 jΦ(x; y)j ≤ kkΦk jxj dµ1 jyj dµ2 K1 K2 for x 2 C(K1) and y 2 C(K2). The smallest constant k in the above theorem is called the C Grothendieck constant, denoted KG . We have known C 4/π ≤ KG < 1:405. Therefore, the constant k in the theorem may be chosen independent of the spaces K1 and K2. Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 7 / 23 Proof. ∗ We note that (C(K1)⊗^ C(K2)) = BL(C(K1); C(K2); C). n Z Z 1=2 C X 2 2 kukp = sup jΦ(u)j ≤ KG jxi j dµ1 jyi j dµ2 Φ2[BL(C(K1);C(K2);C)]1 i=1 Z n Z n ! n n ! X 2 X 2 X 2 X 2 ≤ c jxi j dµ1 + jyi j dµ2 ≤ c k jxi j k + k jyi j k : i=1 i=1 i=1 i=1 As a consequence of the Grothendieck Theorem we have Corollary 2 1 C Let K1; K2 be compact Hausdorff spaces and c = 2 KG . Then for each Pn u = i=1 xi (t) ⊗ yi (t) 2 C(K1) ⊗ C(K2) we have n n ! X 2 X 2 kukp ≤ c k jxi (t)j k1 + k jyi (t)j k1 : i=1 i=1 Y. Zhang (U of Manitoba) Banach algebra valued continuous functions 8 / 23 As a consequence of the Grothendieck Theorem we have Corollary 2 1 C Let K1; K2 be compact Hausdorff spaces and c = 2 KG . Then for each Pn u = i=1 xi (t) ⊗ yi (t) 2 C(K1) ⊗ C(K2) we have n n ! X 2 X 2 kukp ≤ c k jxi (t)j k1 + k jyi (t)j k1 : i=1 i=1 Proof. ∗ We note that (C(K1)⊗^ C(K2)) = BL(C(K1); C(K2); C). n Z Z 1=2 C X 2 2 kukp = sup jΦ(u)j ≤ KG jxi j dµ1 jyi j dµ2 Φ2[BL(C(K1);C(K2);C)]1 i=1 Z n Z n ! n n ! X 2 X 2 X 2 X 2 ≤ c jxi j dµ1 + jyi j dµ2 ≤ c k jxi j k + k jyi j k : i=1 i=1 i=1 i=1 Y.