The Dauns-Hofmann Theorem and tensor products of C*-algebras

22 October 2013
David McConnell
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.
  • Functional Analysis Seminar