On Banach space projective tensor product of $C^*$-algebras

TL;DR

This paper investigates the algebraic and structural properties of the Banach space projective tensor product of $C^*$-algebras, revealing its injectivity, ideal structure, and center identification, and comparing it with known tensor products.

Contribution

It provides new insights into the structure of the Banach space projective tensor product of $C^*$-algebras, including ideal structure and center characterization, extending previous understanding.

Findings

Injectivity of the tensor product on $C^*$-algebras

Detailed description of closed ideals in the tensor product

Identification of the center of the tensor product as $Z(A) ensor_{g} Z(B)

Abstract

We analyze certain algebraic structures of the Banach space projective tensor product of $C^*$-algebras which are comparable with their known counterparts or the Haagerup tensor product and the operator space projective tensor product of $C^*$-algebras. Highlights of this analysis include (a) injectivity of the Banach space projective tensor product when restricted to the tensor products of $C^*$-algebras, (b) detailed structure of closed ideals of $A \otimes_{\gamma} B$ in terms of those of $A$ and $B$, (c) identification of certain spaces of ideals of $A \otimes_{\gamma} B$ in terms of those of $A$ and $B$ from the perspective of hull-kernel topology, and (d) identification of the center of $A \otimes_{\gamma} B$ with $Z(A) \otimes_{\gamma} Z(B)$, where $A$ and $B$ are $C^*$-algebras.