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The equality of the reduced and the full \(C^*\)-algebras and the amenability of a topological groupoid. (English) Zbl 1062.22004

Gaşpar, D. (ed.) et al., Recent advances in operator theory, operator algebras, and their applications. Proceedings of the 19th international conference on operator theory (OT 19), Timişoara, Romania, June 27–July 2, 2002. Basel: Birkhäuser (ISBN 3-7643-7127-7/hbk). Operator Theory: Advances and Applications 153, 61-78 (2005).
Summary: C. Anantharaman-Delaroche and J. Renault [Amenable groupoids (Geneva 2000; Zbl 0960.43003)] have proved that the amenability of a topological locally compact groupoid implies the equality of the reduced and the full \(C^*\)-algebras. In this paper we shall prove the converse assertion under a technical hypothesis. We shall prove that if \(G\) is a locally compact second countable groupoid endowed with a Haar system having “a bounded decomposition over the principal groupoid associated to \(G\)”, then the equality \(C^*_{\text{red}}(G)= C^*(G)\) implies the amenability of all quasi-invariant measures. In order to prove this we shall see that the inequality \(\|\text{II}_\mu(f)\|\leq \|\text{Reg}_\mu(f)\|\) for all \(f\in C_c(G)\) implies a similar inequality for all \(f\in I(G,\nu,\mu)\) (where \(\text{Reg}_\mu\) is the left regular representation of \(C_c(G)\) on a quasi-invariant measure \(\mu\), and \(\text{II}\mu\) is the trivial representation on \(\mu\)).
For the entire collection see [Zbl 1051.46002].

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
43A07 Means on groups, semigroups, etc.; amenable groups
46L52 Noncommutative function spaces
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)

Citations:

Zbl 0960.43003
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