𝐶*-algebras of right LCM monoids and their equilibrium states

Abstract

We study the internal structure of C * -algebras of right LCM monoids by means of isolating the core semigroup C * -algebra as the coefficient algebra of a Fock-type module on which the full semigroup C * -algebra admits a left action. If the semigroup has a generalised scale, we classify the KMS-states for the associated time evolution on the semigroup C * -algebra, and provide sufficient conditions for uniqueness of the KMS β -state at inverse temperature β in a critical interval.

A von Neumann algebraic approach to self-similar group actions

“…for any g 1 , g 2 ∈ G. Repeating the same argument several times, we have µ( g∈F (X ω ) g-gen ) = 1 for any finite set F ⊂ G. Since G is countable, we get the equivalence of (1) and (2). Similarly, (3) and (4) are equivalent.…”

“…Assuming µ((X ω ) G-gen ) = 1, we observe that there exist unique KMS states on O Gmax and O Gmin for the canonical gauge action. The existence and uniqueness of the KMS state on O Gmax is already proved in [9] for contracting cases and in [2] for more general cases (the argument in [2] is not restricted to self-similar group actions). In this paper, we show the existence of the KMS state on O Gmin .…”

“…Actually, there exists a unique pre-KMS function on G in case µ((X ω ) G-gen ) = 1. The following proposition was already proved in [2,Proposition 8.3] but for the reader's convenience we prove here without using more general terminology in [2]. Proposition 3.8.…”

“…We also have the following theorem as a corollary of the above theorem. However, the following theorem was already proved in [2] from arguments on KMS states on Toeplitz type C * -algebra associated to right LCM monoids.…”

On the Simplicity of C$^*$-algebras Associated to Multispinal Groups

“…Define a function ψ on G, X by ψ(S u gS * v ) = δ u,v µ((X ω ) g ) Here δ is the Kronecker delta. If G is contracting, then the function ψ extends to KMS states ψ min and ψ max on O Gmin and O Gmax , respectively [7,16,23]. Here note that we have canonical embeddings of G into O Gmin and O Gmax .…”