KMS states on Nica-Toeplitz algebras of product systems

TL;DR

This paper studies KMS states on Nica-Toeplitz algebras of product systems over semigroups, introducing finite type systems and constructing KMS states from traces, extending previous results for affine semigroups.

Contribution

It introduces the concept of product systems of finite type and constructs KMS states for these systems, generalizing earlier work on affine semigroups.

Findings

KMS states are characterized via restrictions to core algebras with scaling conditions.

Construction of KMS states from traces on the coefficient algebra.

Extension of known results to more general product systems and semigroups.

Abstract

We investigate KMS states of Fowler's Nica-Toeplitz algebra $\mathcal{NT}(X)$ associated to a compactly aligned product system $X$ over a semigroup $P$ of Hilbert bimodules. This analysis relies on restrictions of these states to the core algebra which satisfy appropriate scaling conditions. The concept of product system of finite type is introduced. If $(G, P)$ is a lattice ordered group and $X$ is a product system of finite type over $P$ satisfying certain coherence properties, we construct KMS$_\beta$ states of $\NT(X)$ associated to a scalar dynamics from traces on the coefficient algebra of the product system. Our results were motivated by, and generalize some of the results of Laca and Raeburn obtained for the Toeplitz algebra of the affine semigroup over the natural numbers.