KMS states for generalized gauge actions on C*-algebras associated with self-similar sets

Publication date: 7 September 2021

Abstract: Given a self-similar

K

set defined from an iterated function system

G a m m a = (g a m m a_{1}, l d o t s, g a m m a_{n})

and a set of function

H = {h_{i} : K o m a t h b b R}_{i = 1}^{d}

satisfying suitable conditions, we define a generalized gauge action on Kawjiwara-Watatani algebras

m a t h c a l O_{G} a m m a

and their Toeplitz extensions

m a t h c a l T_{G} a m m a

. We then characterize the KMS states for this action. For each

, there is a Ruelle operator

and the existence of KMS states at inverse temperature

is related to this operator. The critical inverse temperature

is such that

has spectral radius 1. If

, there are no KMS states on

m a t h c a l O_{G} a m m a

and

m a t h c a l T_{G} a m m a

; if

, there is a unique KMS state on

m a t h c a l O_{G} a m m a

and

m a t h c a l T_{G} a m m a

which is given by the eigenmeasure of

; and if

, including

, the extreme points of the set of KMS states on

m a t h c a l T_{G} a m m a

are parametrized by the elements of

K

and on

m a t h c a l O_{G} a m m a

by the set of branched points.

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