# The Glimm space of the minimal tensor product of C*-algebras

### Dave McConnell (Trinity College Dublin)

Tuesday 22nd January, 2013 16:00-17:00 Maths 203

#### Abstract

We consider a C*-algebra A and its space of Primitive ideals Prim(A)
in its usual hull-kernel topology.  The Glimm space of A, Glimm(A), is
defined as the complete regularisation of Prim(A) - the quotient of
Prim(A) modulo the relation of inseparability by continuous functions,
with topology induced by the continuous functions on Prim(A).  To each
such equivalence class in Prim(A) we assign a (closed, two-sided)
ideal of A given by the intersection of the primitive ideals contained
in it. Thus Glimm(A) may be regarded as a set of ideals of A.

For the minimal tensor product A \otimes B of C*-algebras A and B we
describe Glimm (A \otimes B ) in terms of Glimm(A) and Glimm(B), both
topologically and as a set of ideals of A \otimes B, generalising a
result of Kaniuth (1994). We apply this to the study of the centre of
the multiplier algebra of A \otimes B.