Ideals in Operator Space Projective Tensor Product of $C^{\ast }$-algebras

Abstract

For $C^{\ast }$-algebras $A$ and $B$, we prove the slice map conjecture for ideals in the operator space projective tensor product $A\hat{\otimes }B$. As an application, a characterization of prime ideals in the Banach $\ast$-algebra $A\hat{\otimes }B$ is obtained. Further, we study the primitive ideals, modular ideals and the maximal modular ideals of $A\hat{\otimes }B$. It is also shown that the Banach $\ast$-algebra $A\hat{\otimes }B$ possesses Wiener property; and that, for a subhomogenous $C^{\ast }$-algebra $A$, $A\hat{\otimes }B$ is symmetric.

11 pages