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Register a new userKMS states for the generalized gauge action on graph algebras
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Given a positive function on the set of edges of an arbitrary directed graph $E=(E^0,E^1)$, we define a one-parameter group of automorphisms on the C*-algebra of the graph $C^*(E)$, and study the problem of finding KMS states for this action. We prove that there are bijective correspondences between KMS states on $C^*(E)$, a certain class of states on its core, and a certain class of tracial states on $C_0(E^0)$. We also find the ground states for this action and give some examples.
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For a finite, strongly connected $k$-graph $Lambda$, an Huef, Laca, Raeburn and Sims studied the KMS states associated to the preferred dynamics of the $k$-graph $C^*$-algebra $C^*(Lambda)$. They found that these KMS states are determined by the periodicity of $Lambda$ and a certain Borel probability measure $M$ on the infinite path space $Lambda^infty$ of $Lambda$. Here we consider different dynamics on $C^*(Lambda)$, which arise from a functor $y: Lambda to mathbb{R}_+$ and were first proposed by McNamara in his thesis. We show that the KMS states associated to McNamaras dynamics are again parametrized by the periodicity group of $Lambda$ and a family of Borel probability measures on the infinite path space. Indeed, these measures also arise as Hausdorff measures on $Lambda^infty$, and the associated Hausdorff dimension is intimately linked to the inverse temperatures at which KMS states exist. Our construction of the metrics underlying the Hausdorff structure uses the functors $y: Lambda to mathbb{R}_+$; the stationary $k$-Bratteli diagram associated to $Lambda$; and the concept of exponentially self-similar weights on Bratteli diagrams.
Given a self-similar $K$ set defined from an iterated function system $Gamma=(gamma_1,ldots,gamma_n)$ and a set of function $H={h_i:Ktomathbb{R}}_{i=1}^d$ satisfying suitable conditions, we define a generalized gauge action on Kawjiwara-Watatani algebras $mathcal{O}_Gamma$ and their Toeplitz extensions $mathcal{T}_Gamma$. We then characterize the KMS states for this action. For each $betain(0,infty)$, there is a Ruelle operator $mathcal{L}_{H,beta}$ and the existence of KMS states at inverse temperature $beta$ is related to this operator. The critical inverse temperature $beta_c$ is such that $mathcal{L}_{H,beta_c}$ has spectral radius 1. If $beta<beta_c$, there are no KMS states on $mathcal{O}_Gamma$ and $mathcal{T}_Gamma$; if $beta=beta_c$, there is a unique KMS state on $mathcal{O}_Gamma$ and $mathcal{T}_Gamma$ which is given by the eigenmeasure of $mathcal{L}_{H,beta_c}$; and if $beta>beta_c$, including $beta=infty$, the extreme points of the set of KMS states on $mathcal{T}_Gamma$ are parametrized by the elements of $K$ and on $mathcal{O}_Gamma$ by the set of branched points.
We extend a result about the gauge action on noncommutative solitons by showing that a family of functions can be gauged away to a Gaussian using the quantification condition given in On a gauge action on sigma model solitons IDAQP(2018).
Recently, examples of an index theory for KMS states of circle actions were discovered, cite{CPR2,CRT}. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C*-algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in cite{CPR2,CRT} as special cases. Next we use the Araki-Woods III_lambda representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems.
It is shown that any bundle of KMS state spaces which can occur for a flow on a unital separable C*-algebra with a trace state can also be realized by a flow on any given unital infinite-dimensional simple AF algebra.
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