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Authors
Gabor Szabo
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In this paper we consider a bootstrap class $mathfrak C$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on C*-algebras. This class includes all torsion-free abelian groups, poly-$mathbb Z$-groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $Gammainmathfrak C$ and any strongly self-absorbing C*-algebra $mathcal D$: (1) There is a unique strongly outer $Gamma$-action on $mathcal D$ up to (very strong) cocycle conjugacy. (2) If $alpha: Gammacurvearrowright A$ is a strongly outer action on a separable, unital, nuclear, simple, $mathcal D$-stable C*-algebra with at most one trace, then it absorbs every $Gamma$-action on $mathcal D$ up to (very strong) cocycle conjugacy. In fact we establish more general relati
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This note provides some technical support to the proof of a result of W. Winter which shows that two unital separable simple amenable ${cal Z}$-absorbing C*-algebras with locally finite decomposition property satisfying the UCT whose projections separate the traces are isomorphic if their $K$-theory is finitely generated and their Elliott invariants are the same.
In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly self-absorbing C*-dynamical systems, which was introduced and studied in previous work. In particular, this concerns the theory when restricted to the case where all the semi-strongly self-absorbing actions are assumed to be unitarily regular, which is a mild technical condition. The central result in the first part is a strengthened version of the equivariant McDuff-type theorem, where equivariant tensorial absorption can be achieved with respect to so-called very strong cocycle conjugacy. Secondly, we establish completely new results within the theory. This mainly concerns how equivariantly $cal Z$-stable absorption can be reduced to equivariantly UHF-stable absorption with respect to a given semi-strongly self-absorbing action. Combining these abstract results with known uniqueness theorems due to Matui and Izumi-Matui, we obtain the following main result. If $G$ is a torsion-free abelian group and $cal D$ is one of the known strongly self-absorbing C*-algebras, then strongly outer $G$-actions on $cal D$ are unique up to (very strong) cocycle conjugacy. This is new even for $mathbb{Z}^3$-actions on the Jiang-Su algebra.
The spectral functor of an ergodic action of a compact quantum group G on a unital C*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor *-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C*-algebra. As an application, we associate an ergodic G-action on a unital C*-algebra to an inclusion of Rep(G) into an abstract tensor C*-category. If the inclusion arises from a quantum subgroup of G, the associated G-system is just the quantum quotient space. If G is a group and the category has permutation symmetry, the associated system is commutative, and therefore isomorphic to the classical quotient space by a closed subgroup of $G$. If a tensor C*-category has a Hecke symmetry making an object of dimension d and q-quantum determinant one then there is an ergodic action of S_qU(d) on a unital C*-algebra, having the spaces of intertwiners from the tensor unit to powers of the object as its spectral subspaces. The special case od S_qU(2) is discussed.
We define a notion of tracial $mathcal{Z}$-absorption for simple not necessarily unital C*-algebras. This extends the notion defined by Hirshberg and Orovitz for unital (simple) C*-algebras. We provide examples which show that tracially $mathcal{Z}$-absorbing C*-algebras need not be $mathcal{Z}$-absorbing. We show that tracial $mathcal{Z}$-absorption passes to hereditary C*-subalgebras, direct limits, matrix algebras, minimal tensor products with arbitrary simple C*-algebras. We find sufficient conditions for a simple, separable, tracially $mathcal{Z}$-absorbing C*-algebra to be $mathcal{Z}$-absorbing. We also study the Cuntz semigroup of a simple tracially $mathcal{Z}$-absorbing C*-algebra and prove that it is almost unperforated and weakly almost divisible.
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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