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

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).