The representation theory of C*-algebras associated to groupoids

Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of such that G: = E/ is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E; G, λ) is a quotient of the C*-algebra of E obtained by completing the space of -equivariant functions on E. We show that C*(E; G, λ) is postliminal if and only if the orbit space of G is T0 and that C*(E; G, λ) is liminal if and only if the orbit space is T1. We also show that C*(E; G, λ) has bounded trace if and only if G is integrable and that C*(E; G, λ) is a Fell algebra if and only if G is Cartan.Let be a second-countable, locally compact, Hausdorff groupoid with Haar system λ and continuously varying, abelian isotropy groups. Let be the isotropy groupoid and : = /. Using the results about twisted groupoid C*-algebras, we show that the C*-algebra C*(, λ) has bounded trace if and only if is integrable and that C*(, λ) is a Fell algebra if and only if is Cartan. We illustrate our theorems with examples of groupoids associated to directed graphs.

Spectrally bounded traces on C*-algebras

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant M ≥ 0 such that r(T x) ≤ Mr(x) for all x ∈ E, where r (·) denotes the spectral radius. We establish the equivalence of the following properties of a unital linear mapping T from a unital C* -algebra A into its centre:(a) T is spectrally bounded;(b) T is a spectrally bounded trace;(c) T is a bounded trace.

GROUPS WITH BOUNDED TRACE AND THE TRIVIAL CORTEX PROBLEM

It will be shown that a connected Lie group has bounded trace if and only if its component with no semisimple factors has a cocompact radical which is a direct product E × R where E denotes a finite direct product of groups of motion of the real plane (including covering groups) and R is a semidirect product of a torus and a nilpotent group with continuous trace. This leads to a complete description of the class of all connected Lie groups whose identity representation constitutes a separated point in the unitary dual ( Cor (G) = {1G}).