On homomorphisms of matrix algebras of continuous functions

Marius Dadarlat

Vol. 132 (1988), No. 2, 227–231

DOI: 10.2140/pjm.1988.132.227

Abstract

If X is a topological space we denote by C(X) ⊗M_n the algebra of continuous functions from X to the algebra M_n of n × n complex matrices. A complete characterization of those topological spaces Y is given (in terms of vector bundles on Y ) such that each unital algebra-homomorphism Φ : C(X) ⊗M_n → C(Y ) ⊗M_kn is of the form α∘ (Φ′⊗ id_n) for some homomorphism Φ′ : C(X) → C(Y ) ⊗ M_k and some suitable inner (or C(Y )-linear) automorphism α of the algebra C(Y ) ⊗ M_kn. In particular this decomposition is assured provided that Y is a finite CW-complex of dimension ≤ 2k and K⁰(Y ) does not have n-torsion.

Mathematical Subject Classification 2000

Primary: 46L99

Secondary: 46M05

Milestones

Received: 15 December 1986

Published: 1 April 1988