22 October 2013

17:00

to

18:25

David McConnell

Abstract

The problem of representing a (non-commutative) C*-algebra $A$ as the
algebra of sections of a bundle of C*-algebras over a suitable base
space may be viewed as that of finding a non-commutative Gelfand-Naimark
theorem. The space $\mathrm{Prim}(A)$ of primitive ideals of $A$, with
its hull-kernel topology, arises as a natural candidate for the base
space. One major obstacle is that $\mathrm{Prim}(A)$ is rarely
sufficiently well-behaved as a topological space for this purpose. A
theorem of Dauns and Hofmann shows that any C*-algebra $A$ may be
represented as the section algebra of a C*-bundle over the complete
regularisation of $\mathrm{Prim}(A)$, which is identified in a natural
way with a space of ideals known as the Glimm ideals of $A$, denoted
$\mathrm{Glimm}(A)$.
In the case of the minimal tensor product $A \otimes B$ of two
C*-algebras $A$ and $B$, we show how $\mathrm{Glimm}(A \otimes B)$ may
be constructed in terms of $\mathrm{Glimm}(A)$ and $\mathrm{Glimm} (B)$.
As a consequence, we describe the associated C*-bundle representation of
$A \otimes B$ over this space, and discuss properties of this bundle
where exactness of $A$ plays a decisive role.