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).