Spectral synthesis for operator space projective tensor product of $C^*$-algebras

TL;DR

This paper investigates spectral synthesis in the operator space projective tensor product of $C^*$-algebras, establishing conditions under which spectral synthesis holds, especially when one algebra has finitely many closed ideals.

Contribution

It demonstrates that spectral synthesis is valid for the tensor product if either algebra has finitely many closed ideals, and explores properties of $A ext{op} A$ with reverse involution.

Findings

Spectral synthesis holds when $A$ or $B$ has finitely many closed ideals.

The structure of $A ext{op} A$ with reverse involution is analyzed.

Conditions for spectral synthesis in the tensor product are established.

Abstract

We study the spectral synthesis for the Banach *-algebra $A\oop B$, the operator space projective tensor product of $C^*$-algebras $A$ and $B$. It is shown that if $A$ or $B$ has finitely many closed ideals, then $A\oop B$ obeys spectral synthesis. The Banach algebra $A \oop A$ with the reverse involution is also studied.