Traces on locally compact groups
Brian Forrest, Nico Spronk, Matthew Wiersma

TL;DR
This paper systematically studies traces on locally compact groups and their C*-algebras, introducing the trace kernel and residually-SIN groups, and explores implications for amenability, property (T), and embeddability into AF algebras.
Contribution
It introduces the class of residually-SIN groups, analyzes trace kernels for various groups, and links traces to properties like amenability, property (T), and embeddability into simple AF algebras.
Findings
Existence of traces on reduced C*-algebras is characterized for compactly generated groups.
Unique traces are shown to exist on certain non-discrete groups.
Amenability is equivalent to nuclearity and trace existence in reduced C*-algebras.
Abstract
We conduct a systematic study of traces on locally compact groups, in particular traces on their universal and reduced C*-algebras. We introduce the trace kernel, and examine its relation to the von Neumann kernel and to small-invariant neighbourhood (SIN) quotients. In doing so, we introduce the class of residually- groups, which contains both and maximally almost periodic groups. We examine in detail the trace kernel for connected groups. We study traces on reduced C*-algebras, giving a simple proof for compactly generated groups that existence of such a trace is equivalent to having an open normal amenable subgroup, and we display non-discrete groups admitting unique trace. We finish by examining amenable traces and the factorization property. We show for property (T) groups that amenable trace kernels coincide with von Neumann kernels. We show for totally disconnected…
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Taxonomy
TopicsAdvanced Operator Algebra Research · Spectral Theory in Mathematical Physics
