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On homomorphisms of matrix algebras of continuous functions. (English) Zbl 0652.46040
We quote a result from this rather technical paper. Denote by \(Vect_ m(Y)\) the set of isomorphism classes of complex vector bundles of rank m on a topological space Y. Let \(T_ nVect_ m(Y)\) be the subset of \(Vect_ m(Y)\) given by all vector bundles E such that the direct sum \(E\oplus E\oplus...\oplus E\) (n-times) is isomorphic to the trivial bundle of rank nm. Denote by \(C(Y)\oplus M_ n\) the algebra of all continuous functions from Y to \(n\times n\) complex matrices.
Theorem 2. Let X, Y be topological spaces. Then the following assertions are equivalent:
(i) The set \(T_ nVect_ k(Y)\) reduces to the trivial bundle of rank k.
(ii) Each homomorphism \(\Phi \in Hom(C(X)\otimes M_ n,C(Y)\otimes M_{kn})\) is inner equivalent to a homomorphism of the form \(\Phi'\otimes id_ n\) for some \(\Phi'\in Hom(C(X),C(Y)\otimes M_ k)\).