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Register a new userLocalizing the Elliott Conjecture at Strongly Self-absorbing C*-algebras --An Appendix
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Huaxin Lin
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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.
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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.
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
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.
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.
Given a graph $E$, an action of a group $G$ on $E$, and a $G$-valued cocycle $phi$ on the edges of $E$, we define a C*-algebra denoted ${cal O}_{G,E}$, which is shown to be isomorphic to the tight C*-algebra associated to a certain inverse semigroup $S_{G,E}$ built naturally from the triple $(G,E,phi)$. As a tight C*-algebra, ${cal O}_{G,E}$ is also isomorphic to the full C*-algebra of a naturally occurring groupoid ${cal G}_{tight}(S_{G,E})$. We then study the relationship between properties of the action, of the groupoid and of the C*-algebra, with an emphasis on situations in which ${cal O}_{G,E}$ is a Kirchberg algebra. Our main applications are to Katsura algebras and to certain algebras constructed by Nekrashevych from self-similar groups. These two classes of C*-algebras are shown to be special cases of our ${cal O}_{G,E}$, and many of their known properties are shown to follow from our general theory.
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