Abstract. For a simple C * -algebra A and any other C * -algebra B, it is proved that every closed ideal of A ⊗ min B is a product ideal if either A is exact or B is nuclear.

Let A and B be C * -algebras. We prove the slice map conjecture for ideals in the operator space projective tensor product A ⊗ B. As an application, a characterization of the prime ideals in the Banach * -algebra A ⊗ B is obtained. In addition, we study the primitive ideals, modular ideals and the maximal modular ideals of A ⊗ B. We also show that the Banach * -algebra A ⊗ B possesses the Wiener property and that, for a subhomogeneous C * -algebra A, the Banach * -algebra A ⊗ B is symmetric.2010 Mathematics subject classification: primary 46L06; secondary 46L07, 47L25.

For C * -algebras A and B, the identity map from A ⊗B into A⊗ λ B is shown to be injective. Next, we deduce that the center of the completion of the tensor product A ⊗ B of two C * -algebras A and B with centers Z (A) and Z (B) under operator space projective norm is equal to Z (A) ⊗Z (B). A characterization of isometric automorphisms of A ⊗B and A⊗ h B is also obtained.

No abstract

We prove that, for operator spaces V and W , the operator space V * * ⊗ h W * * can be completely isometrically embedded into (V ⊗ h W ) * * , ⊗ h being the Haagerup tensor product. We also show that, for exact operator spaces V and W , a jointly completely bounded bilinear form on V × W can be extended uniquely to a separately w * -continuous jointly completely bounded bilinear form on V * * ×W * * . This paves the way to obtaining a canonical embedding of V * * ⊗ W * * into (V ⊗ W ) * * with a continuous inverse, where ⊗ is the operator space projective tensor product. Further, for C * -algebras A and B, we study the (closed) ideal structure of A ⊗ B, which, in particular, determines the lattice of closed ideals of B(H) ⊗ B(H) completely.