Ideals in tensor products, topological dimension zero and the weak ideal property

Abstract

For any property (P) of C*-algebras that passes to ideals and admits largest ideals we characterize when the largest ideal in a tensor product of two C*-algebras which has (P) is the tensor product of the largest ideals in the tensor factors which have (P), whenever one of the tensor factors is exact. We prove that we have this equality, for separable C*-algebras, when (P) is topological dimension zero. We give new characterizations of topological dimension zero, the weak ideal property, pure infiniteness and strong pure infiniteness (in fact, we prove a more general result.) Let I be an ideal of A⊗B, where A and B are non-zero C*-algebras and A or B is exact. We associate to I two canonical ideals, IA (see also [1]) and I(A) of A, and two canonical ideals, IB and I(B) of B. We study the interplay between I and these four ideals, and we characterize (under some mild assumptions) several interesting and natural conditions on I in terms of IA, I(A), IB and I(B).