The Glimm space of the minimal tensor product of C$^{\ast}$-algebras

TL;DR

This paper explores the structure of the Glimm space of the minimal tensor product of C*-algebras, establishing a natural bijection and analyzing the continuity and ideal-related properties of this construction.

Contribution

It introduces a natural open bijection between the Glimm spaces of tensor products and identifies conditions for continuity and ideal structure in this context.

Findings

Established a natural open bijection between Glimm(A)  Glimm(B) and Glimm(A  B)

Determined the structure space of the center of the multiplier algebra of the tensor product

Provided necessary and sufficient conditions for the surjectivity of the inclusion ZM(A)  ZM(B)  ZM(A  B)

Abstract

We show that for C$^{\ast}$-algebras $A$ and $B$, there is a natural open bijection from $\mathrm{Glimm}(A) \times \mathrm{Glimm} (B)$ to $\mathrm{Glimm}(A \otimes_{\alpha} B)$ (where $A \otimes_{\alpha} B$ denotes the minimal C$^{\ast}$-tensor product), and identify a large class of C$^{\ast}$-algebras $A$ for which the map is continuous for arbitrary $B$. As a consequence we determine the structure space of the centre of the multiplier algebra $ZM(A \otimes_{\alpha} B)$ in terms of $\mathrm{Glimm}(A)$ and $\mathrm{Glimm} (B)$, and give necessary and sufficient conditions for the inclusion $ZM(A) \otimes ZM(B) \subseteq ZM(A \otimes_{\alpha} B)$ to be surjective. Further we show that when the Glimm spaces are considered as sets of ideals, the map $(G,H) \mapsto G \otimes_{\alpha} B + A \otimes_{\alpha} H$ implements the above bijection, extending a result of Kaniuth from a 1996 paper by eliminating the assumption of property (F).