Weak Centrality for Certain Tensor Products of $C^\ast$-algebras

TL;DR

This paper investigates the weak centrality properties of specific tensor products of $C^*$-algebras, establishing how these properties relate to the factors and identifying maximal weakly central ideals.

Contribution

It provides new insights into the weak centrality of tensor products of $C^*$-algebras, especially for Haagerup and Banach space projective tensor products, and identifies their largest weakly central ideals.

Findings

Weak centrality of tensor products depends on the factors' properties.

Largest weakly central ideal can be explicitly identified in certain cases.

Discussion of centrality and quasi-centrality of these tensor products.

Abstract

In this article, we discuss the weak centrality of the tensor product $A\otimes_\alpha B$ of $C^\ast$-algebras $A$ and $B$ in terms of the weak centrality of $A$ and $B$, where $\alpha$ is either the Haagerup or the Banach space projective tensor product. In the due course, we also identify the largest weakly central ideal of $A\otimes_\alpha B$ in certain cases. Centralilty and quasi-centrality of these tensor products are also discussed.