Ideals in Operator Space Projective Tensor Product of $C^*$-algebras

TL;DR

This paper proves the slice map conjecture for ideals in the operator space projective tensor product of $C^*$-algebras, providing new characterizations and properties of ideals in this tensor product space.

Contribution

It establishes the slice map conjecture for ideals in the operator space projective tensor product of $C^*$-algebras and explores the structure of various types of ideals.

Findings

Proved the slice map conjecture for ideals in $A \u2208 ext{tensor} B$.

Characterized prime ideals in $A \u2208 ext{tensor} B$.

Showed $A \u2208 ext{tensor} B$ has Wiener property and is symmetric for subhomogeneous $A$.

Abstract

For $C^*$-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 $C^*$-algebra $A$, $A\hat\otimes B$ is symmetric.