𝐶*-algebras with Hausdorff spectrum

Bunce, John W.; Deddens, James A.

doi:10.1090/S0002-9947-1975-0405116-1

$C^*$-algebras with Hausdorff spectrum
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by John W. Bunce and James A. Deddens PDF

Trans. Amer. Math. Soc. 212 (1975), 199-217 Request permission

Abstract:

By the spectrum of a ${C^\ast }$-algebra we mean the set of unitary equivalence classes of irreducible representations equipped with the hull-kernel topology. We are concerned with characterizing the ${C^\ast }$-algebras with identity which have Hausdorff spectrum. We characterize the ${C^\ast }$-algebras with identity and bounded representation dimension which have Hausdorff spectrum. Our results are more natural when the ${C^\ast }$-algebra is singly generated. For singly generated ${C^\ast }$-algebras with unbounded representation dimension, we reduce the problem to the case when the generator is an infinite direct sum of irreducible finite scalar matrices, and we have partial results in this case.

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