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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   $C^{\ast }$  -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.

KMS States of Quasi-Free Dynamics on C * -Algebras of Product Systems over Right LCM Monoids