A Bitopological Gelfand Theorem for C^*-Algebras
by
John Mack
University of Kentucky

Let E, X, and Y be topological spaces. Then (E, X, Y, \phi) is a field of topological spaces if \phi is an open surjection from X ×Y to E so that there exists a map p : E --> X for which p o \phi is the projection of X ×Y onto X. A section is a map s : X --> E for which p o s is the identity map on X.

Let A be a C^*-algebra with norm topology \eta and Id(A) be the lattice of closed ideals of A. If X is a Lawson closed subset of Id(A) containing Prim(A), let \omega, \sigma and \lambda denote the lower, Scott and Lawson topologies on X. Define E = \cup { I } ×A / I where the union is taken over X. Let \phi: X ×A --> E, be the map given by \phi(I, a) = (I, a+I).

Theorem 1 There exist topologies L, U on E for which (E, L, U) is a pairwise completely regular bitopological space so that \phi is pairwise continuous while (E, X, A, \phi) is a field of topological spaces with respect to the \lambda×\eta topology on X ×A and the L \/ U topology on E.

Theorem 2 Let X = Glimmal(A); then X is an \omega closed subspace of Id(A) that contains Prim(A). Also

1. The normed algebra of all pairwise continuous sections s : X --> E is isometrically isomorphic to A.
2. If A is a commutative C^*-algebra, then X = Max(A) \cup { A } and the representation in part (1) reduces to the standard Gelfand transform.
3. If Prim(A) is Hausdorff for the hull-kernel topology, then X = Prim(A) \cup { A } and the representation in (1) is equivalent to the Fell representation of A as a continuous field of C^*-algebras.

Date received: June 29, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-45.