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Register a new userThe bundle of KMS state spaces for flows on a unital C*-algebra
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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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We define a groupoid from a labelled space and show that it is isomorphic to the tight groupoid arising from an inverse semigroup associated with the labelled space. We then define a local homeomorphism on the tight spectrum that is a generalization of the shift map for graphs, and show that the defined groupoid is isomorphic to the Renault-Deaconu groupoid for this local homeomorphism. Finally, we show that the C*-algebra of this groupoid is isomorphic to the C*-algebra of the labelled space as introduced by Bates and Pask.
In this article, we give an abstract characterization of the ``identity of an operator space $V$ by looking at a quantity $n_{cb}(V,u)$ which is defined in analogue to a well-known quantity in Banach space theory. More precisely, we show that there exists a complete isometry from $V$ to some $mathcal{L}(H)$ sending $u$ to ${rm id}_H$ if and only if $n_{cb}(V,u) =1$. We will use it to give an abstract characterization of operator systems. Moreover, we will show that if $V$ is a unital operator space and $W$ is a proper complete $M$-ideal, then $V/W$ is also a unital operator space. As a consequece, the quotient of an operator system by a proper complete $M$-ideal is again an operator system. In the appendix, we will also give an abstract characterisation of ``non-unital operator systems using an idea arose from the definition of $n_{cb}(V,u)$.
We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for alpha. We extend Exels construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.
We investigate the notion of tracial $mathcal Z$-stability beyond unital C*-algebras, and we prove that this notion is equivalent to $mathcal Z$-stability in the class of separable simple nuclear C*-algebras.
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.
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