The groupoid algebra of an eigenvalue pattern

The eigenvalue pattern of a *-homomorphism between two matrix algebras over commutative C-algebras is a generalization of the Gelfand map in the commutative case. We give a systematic formulation of abstract eigenvalue pattern and extend the classical results by using a technique involving the groupoid algebras of eigenvalue patterns. In the case with matrix algebras over the one-dimensional circle, we characterize all the *-homomorphisms up to unitary equivalence by their eigenvalue patterns. Moreover, this technique has an application to recent classification theorems of C-algebras proved by the present author. Copyright © 2006 American Mathematical Society.