2022/05/22 by Luca Eva Gazdag, Gazdag, Luca Eva, Marcelo Laca +3
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2205.10786
openalex publication_date 2022/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.