The noncommutative topology of one-dimensional spaces

Terry Atherton Loring

Vol. 136 (1989), No. 1, 145–158

DOI: 10.2140/pjm.1989.136.145

Abstract

If X and Y are compact topological spaces, the unital star-homomorphisms from C(X) to C(Y ) satisfy certain homotopy properties when X is an absolute neighborhood retract. We show that two of these properties still hold when C(Y ) is replaced by a “noncommutative space”, i.e. an arbitrary unital C^∗-algebra, but only under the additional assumption that X is one-dimensional. Specifically, we show that C(X) is semiprojective and that two unital star-homomorphisms from C(X) to a C^∗-algebra A are homotopic whenever they are close.

Mathematical Subject Classification 2000

Primary: 46L99

Secondary: 46L05, 55P55

Milestones

Received: 25 August 1987

Revised: 28 February 1988

Published: 1 January 1989