Principal groupoid $C^*$-algebras with bounded trace

Lisa Orloff Clark, Astrid an Huef
2007 Proceedings of the American Mathematical Society  
Suppose G is a second countable, locally compact, Hausdorff, principal groupoid with a fixed left Haar system. We define a notion of integrability for groupoids and show G is integrable if and only if the groupoid C * -algebra C * (G) has bounded trace. Let A be a C * -algebra. An element a of the positive cone A + of A is called a bounded-trace element if the map π → tr(π(a)) is bounded on the spectrum of A; the linear span of the bounded-trace elements is a two-sided * -ideal in A. We say A
more » ... s bounded trace if the ideal of (the span of) the bounded-trace elements is dense in A. Throughout, G is a locally compact, Hausdorff groupoid; in our main results G is assumed to be second-countable and principal. We denote the unit space of G by G 0 , and the range and source maps r, s : G → G 0 are r(γ) = γγ −1 and s(γ) = γ −1 γ, respectively. We let π : G → G 0 × G 0 be the map π(γ) = (r(γ), s(γ)); recall that G is principal if π is injective. In order to define the groupoid C * -algebra, we also assume that G is equipped with a fixed left Haar system: a set {λ x : x ∈ G 0 } of non-negative Radon measures on G such that (3) for f ∈ C c (G) and γ ∈ G, the following equation holds: , gives a fixed right Haar system such that the measures are supported on s −1 ({x}) and for f ∈ C c (G) and γ ∈ G. We will move freely between these two Haar systems. If N ⊆ G 0 , then the saturation of N is r(s −1 (N )) = s(r −1 (N )). In particular, we call the saturation of {x} the orbit of x ∈ G 0 and denote it by [x]. If G is principal and all the orbits are locally closed, then by [4, Proposition 5.1] the orbit space G
doi:10.1090/s0002-9939-07-09035-1 fatcat:clbnwuayrrei7b7ecttjgjxqsi