Product systems over Ore monoids

TL;DR

This paper explores the structure of Cuntz-Pimsner algebras over Ore monoids, using inductive limits, Fell bundles, and groupoid models to analyze their properties and invariants.

Contribution

It introduces a new interpretation of the covariance condition, constructs groupoid models for these algebras, and characterizes their effectiveness and invariant measures.

Findings

Cuntz-Pimsner algebras over Ore monoids can be built via inductive limits.

A groupoid model for these algebras is constructed from monoid actions.

Conditions for effectiveness and invariance of the groupoid are characterized.

Abstract

We interpret the Cuntz-Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz-Pimsner algebras over Ore monoids are constructed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz-Pimsner algebra coming from an action of an Ore monoid on a space by topological correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures.