Ideal Spaces of the Haagerup Tensor Product of C*-Algebras

Abstract

Following the work of Allen, Sinclair and Smith on the primitive ideal space of the Haagerup tensor product A ⊗ hB of C*-algebras A and B, we investigate the hull-kernel topology and use this to determine various other ideal spaces and their topologies in relation to the corresponding ideal spaces of A and B. We study the semi-continuity of norm functions I → ||x + I||(x ∈ A ⊗h B) on these ideal spaces and identify the separated points of Prim (A ⊗h B). Finally, we exhibit several conditions each of which is equivalent to the quasi-standardness of A ⊗h B.