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Register a new userPhase transitions on C*-algebras arising from number fields and the generalized Furstenberg conjecture
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In recent work, Cuntz, Deninger and Laca have studied the Toeplitz type C*-algebra associated to the affine monoid of algebraic integers in a number field, under a time evolution determined by the absolute norm. The KMS equilibrium states of their system are parametrized by traces on the C*-algebras of the semidirect products $J rtimes O^*$ resulting from the multiplicative action of the units $O^*$ on integral ideals $J$ representing each ideal class. At each fixed inverse temperature $beta > 2$, the extremal equilibrium states correspond to extremal traces of $C^*(Jrtimes O^*)$. Here we undertake the study of these traces using the transposed action of $O^*$ on the duals $hat J$ of the ideals and the recent characterization of traces on transformation group C*-algebras due to Neshveyev. We show that the extremal traces of $C^*(Jrtimes O^*)$ are parametrized by pairs consisting of an ergodic invariant measure for the action of $O^*$ on $hat{J}$ together with a character of the isotropy subgroup associated to the support of this measure. For every ideal the dual group $hat {J}$ is a d-torus on which $O^*$ acts by linear toral automorphisms. Hence, the problem of classifying all extremal traces is a generalized version of Furstenbergs celebrated $times 2$ $times 3$ conjecture. We classify the results for various number fields in terms of ideal class group, degree, and unit rank, and we point along the way the trivial, the intractable, and the conjecturally classifiable cases. At the topological level, it is possible to characterize the number fields for which infinite $O^*$-invariant sets are dense in $hat{J} $, thanks to a theorem of Berend; as an application we give a description of the primitive ideal space of $C^*(Jrtimes O^*)$ for those number fields.
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I. Raeburn and J. Taylor have constructed continuous-trace C*-algebras with a prescribed Dixmier-Douady class, which also depend on the choice of an open cover of the spectrum. We study the asymptotic behavior of these algebras with respect to certain refinements of the cover and appropriate extension of cocycles. This leads to the analysis of a limit groupoid G and a cocycle sigma, and the algebra C*(G, sigma) may be regarded as a generalized direct limit of the Raeburn-Taylor algebras. As a special case, all UHF C*-algebras arise from this limit construction.
In this paper we study the C*-envelope of the (non-self-adjoint) tensor algebra associated via subproduct systems to a finite irreducible stochastic matrix $P$. Firstly, we identify the boundary representations of the tensor algebra inside the Toeplitz algebra, also known as its non-commutative Choquet boundary. As an application, we provide examples of C*-envelopes that are not *-isomorphic to either the Toeplitz algebra or the Cuntz-Pimsner algebra. This characterization required a new proof for the fact that the Cuntz-Pimsner algebra associated to $P$ is isomorphic to $C(mathbb{T}, M_d(mathbb{C}))$, filling a gap in a previous paper. We then proceed to classify the C*-envelopes of tensor algebras of stochastic matrices up to *-isomorphism and stable isomorphism, in terms of the underlying matrices. This is accomplished by determining the K-theory of these C*-algebras and by combining this information with results due to Paschke and Salinas in extension theory. This classification is applied to provide a clearer picture of the various C*-envelopes that can land between the Toeplitz and the Cuntz-Pimsner algebras.
In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced $mathrm{C}^*$-algebra $C_r^*(G)$ has real rank zero and strict comparison even though $C_r^*(G)$ may not be nuclear in general. Moreover, if we further assume $G$ being also second countable and non-elementary, then its Cuntz semigroup ${rm Cu}(C_r^*(G))$ is almost divisible and ${rm Cu}(C_r^*(G))$ and ${rm Cu}(C_r^*(G)otimes mathcal{Z})$ are canonically order-isomorphic, where $mathcal{Z}$ denotes the Jiang-Su algebra.
We study subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselters generalization of the Cuntz-Pimsner C*-algebra to the context of subproduct systems. Suppose that $X$ and $Y$ are Arveson-Stinespring subproduct systems associated to two stochastic matrices over a countable set $Omega$, and let $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ be their tensor algebras. We show that every algebraic isomorphism from $mathcal{T}_+(X)$ onto $mathcal{T}_+(Y)$ is automatically bounded. Furthermore, $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ are isometrically isomorphic if and only if $X$ and $Y$ are unitarily isomorphic up to a *-automorphism of $ell^infty(Omega)$. When $Omega$ is finite, we prove that $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ are algebraically isomorphic if and only if there exists a similarity between $X$ and $Y$ up to a *-automorphism of $ell^infty(Omega)$. Moreover, we provide an explicit description of the Cuntz-Pimsner algebra $mathcal{O}(X)$ in the case where $Omega$ is finite and the stochastic matrix is essential.
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.
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