Abstract
Let G be a locally compact groupoid. We recall (Definition 2.2.2) that G is equipped with a left Haar system {λu}.
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Notes
Strictly, a “modular function” needs also to be taken into account.
In the approaches of Renault ([230]) and Muhly ([179]), results of Bourbaki ([19]) and Effros ([82]) respectively are used to facilitate the measure theory. However, in the present (non-Hausdorff) context, it seemed preferable to work out the details from scratch. This also gives a self-contained account, and indeed, as we will see, the details, while requiring care, use only basic measure theory.
The author is grateful to Paul Muhly for helpful discussion about these two proofs of Renault.
Using an easier version of the proof of Theorem 3.1.2, one can also show that the reduced C*red(G), defined below, is isomorphic to K(L2(X, μ)). (Of course, this also follows from the theorem, since C*red(G) is a non-zero homomorphic image of the simple C*-algebraC*(G).)
This is well known but easy to prove directly. Indeed, if f(x,y) = h(x)k(y) = (h⊗k)(x, y) for h, k ∈ Cc(X), then πLtriv (f) is obviously of rank 1, and the span of such operators is norm dense in the algebra of finite rank operators and hence also in the algebra of compact operators. That every πLtriv (f) is compact can be shown by approximating f by linear combinations of functions of the form h⊗k using the Stone-Weierstrass theorem.
In fact, it can be shown that the reduced and universal C*-algebras for Rp are actually isomorphic.
The reader is referred to the work of Nandor Sieben and John Quigg (e.g. [257, 258]) for further advances in the theory of inverse semigroup covariant systems.
Inverse semigroup actions are closely related to the theory of partial actions on C*-algebras developed by Exel and Maclanahan ([94, 171]).
Compatibility is effectively the same as Kumjian’s notion of coherence in [148].
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© 1999 Springer Science+Business Media New York
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Paterson, A.L.T. (1999). Groupoid C*-Algebras and Their Relation to Inverse Semigroup Covariance C*-Algebras. In: Groupoids, Inverse Semigroups, and their Operator Algebras. Progress in Mathematics, vol 170. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1774-9_3
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DOI: https://doi.org/10.1007/978-1-4612-1774-9_3
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