The groupoid algebra of an eigenvalue pattern

Tsang, Kin-Wai

doi:10.1090/S0002-9939-06-08215-3

The groupoid algebra of an eigenvalue pattern
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Proc. Amer. Math. Soc. 134 (2006), 1899-1908 Request permission

Abstract:

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.

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