Separable injectivity and C*-tensor products

Abstract

Let A and B be C*-algebras and let D be a C*-subalgebra of B. We show that if D is separably injective then the triple (A,B,D) verifies the slice map conjecture. As an application, we prove that the minimal C*-tensor product A⊗B is separably injective if and only if both A and B are separably injective and either A or B is finite-dimensional.