Approximating maps and exact C*-algebras

Abstract

Let A and E be C*-algebras, let A ⊗ B denote the minimal C*-tensor product, and let ε A *. The right slice map R: A ⊗ B → B is the unique bounded linear mapping with the property that R (a ⊗ b) = (a)b (a ε A, b ε B)(10). A triple (A, B, D), where D is a C*-subalgebra of B, is said to have the slice map property if whenever x ε A ⊗ B and R(x) D for all ε A* then x ε A ⊗ D). It is known that (A, B, D) has the slice map property whenever A is nuclear (11,13), but it appears to be still unknown whether the nuclearity of B will suffice (unless some extra condition is placed on D (l)).