A Uniqueness Theorem for C*-algebras of Hausdorff Étale Groupoids
Date
2023-04-27
Authors
Goerke, Gavin
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Abstract
In this thesis we study the ideal intersection property for inclusions of C*-algebras C*(H)↪C*(G) induced from a family of open subgroupoids {H} of a locally compact Hausdorff étale groupoid G. For such a family of open subgroupoids we define the notion of relative topological principality and we show that if G is relatively topologically principal to {H} then a representation of C*(G) is faithful if and only if the restriction of the representation to each of the subalgebras C*(H) is faithful. This gives a new method of verifying injectivity of representations of reduced groupoid C*-algebras. As an application of our result we prove a uniqueness theorem for C*-algebras of left cancellative small categories which generalizes a theorem of Marcelo Laca and Camila Sehnem for Toeplitz algebras of group embeddable monoids.
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Keywords
C*-algebras, groupoid, topologically free