KMS states of quasi-free dynamics on $C^*$-algebras of product systems over right LCM monoids

TL;DR

This paper extends the understanding of KMS states for quasi-free dynamics on $C^*$-algebras of product systems over right LCM monoids, providing new positivity conditions and addressing open questions about finite inequality reductions.

Contribution

It generalizes previous results to a broader class of monoids, introduces a combinatorial tree model for analysis, and explores the structure of inverse temperature space.

Findings

Positivity conditions for KMS states are characterized by inequalities.

Finite subset reduction of inequalities is valid for a wider class of monoids.

An example shows a gap in inverse temperature space for certain monoids.

Abstract

We generalise recent results of Afsar, Larsen and Neshveyev for product systems over quasi-lattice orders by showing that the equilibrium states of quasi-free dynamics on the Nica-Toeplitz $C^*$-algebras of product systems over right LCM monoids must satisfy a positivity condition encoded in a system of inequalities satisfied by their restrictions to the coefficient algebra. We prove that the reduction of this positivity condition to a finite subset of inequalities is valid for a wider class of monoids that properly includes finite-type Artin monoids, answering a question left open in their work. Our main technical tool is a combinatorially generated tree modelled on a recent construction developed by Boyu Li for dilations of contractive representations. We also obtain a reduction of the positivity condition to inequalities arising from a certain minimal subset that may not be finite but has the advantage of holding for all Noetherian right LCM monoids, and we present an example, arising from a finite-type Artin monoid, that exhibits a gap in its inverse temperature space.