Strongly zero product determined Banach algebras

Abstract

C*-algebras, group algebras, and the algebra A(X) of approximable operators on a Banach space Xhaving the bounded approximation property are known to be zero product determined. In this paper we give a quantitative estimate of this property by showing that, for the Banach algebra A, there exists a constant awith the property that for every continuous bilinear functional phi: A x A -> C there exists a continuous linear functional xi on A such that

sup(parallel to a parallel to=parallel to b parallel to=1) vertical bar phi(a, b) - xi(ab)vertical bar <= alpha sup(parallel to a parallel to=parallel to b parallel to=1ab-0), vertical bar phi(a, b)vertical bar

in each of the following cases: (i) Ais a C*-algebra, in which case alpha = 8; (ii) A = L-1(G) for a locally compact group G, in which case alpha = 60 root 271+sin pi/10/1-2sin pi/10; (iii) A = A(X) for a Banach space Xhaving property (A)(which is a rather strong approximation property for X), in which case alpha = 60 root 27 1+sin pi/10/1-2sin pi/10 C-2, where C is a constant associated with the property(A) that we require forX.