Sheaves of Measures and KMS-Weights on Topological Graph Algebras

TL;DR

This paper develops a sheaf-theoretic framework for regular Borel measures on topological spaces and applies it to describe KMS-weights on graph C*-algebras using sub-invariant measures.

Contribution

It introduces a sheaf structure for measures and uses it to characterize KMS-weights on topological graph algebras in terms of measures.

Findings

Sheaf structure on measures provides new perspective on measure pullbacks.

KMS-weights are characterized via sub-invariant measures on vertex spaces.

Framework simplifies analysis of KMS-weights in topological graph C*-algebras.

Abstract

We show that the collection of regular Borel measures on a second-countable locally compact Hausdorff space has the structure of a sheaf. With this we give an alternate description of the pullback of a regular Borel measure along a local homeomorphism. We are able to use these tools to give a description of the KMS-weights for the gauge-action on the graph C*-algebra of a second-countable topological graph in terms of sub-invariant measures on the vertex space of said topological graph.