scholarly journals UHF-slicing and classification of nuclear C*-algebras

2014 ◽  
Vol 06 (04) ◽  
pp. 465-540 ◽  
Author(s):  
Karen R. Strung ◽  
Wilhelm Winter
Keyword(s):  
Dynamical Systems ◽  
Classification Theorem ◽  
Discrete Version ◽  
C Algebras ◽  
Minimal Dynamical Systems ◽  
Tracial States

In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0, 1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that such C*-algebras are classifiable by Elliott invariants. Our theorem requires finitely many tracial states that all induce the same state on the K0-group; in particular we do not require that projections separate tracial states. We apply our results to classify some examples of C*-algebras constructed by Elliott to exhaust the invariant. We also give an alternative way to classify examples of Lin and Matui of C*-algebras of minimal dynamical systems. In this way our result can be viewed as a first step towards removing the requirement that projections separate tracial states in the classification theorem for C*-algebras of minimal dynamical systems given by Toms and the second named author.

MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

2011 ◽  
Vol 22 (01) ◽  
pp. 1-23 ◽  
Author(s):  
KAREN R. STRUNG ◽  
WILHELM WINTER
Keyword(s):  
Dynamical Systems ◽  
Compact Metric Space ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Minimal Dynamical Systems ◽  
Uniquely Ergodic ◽  
Stable Classification

Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary that implements α in the transformation group C*-algebra C(X) ⋊α ℤ, and [Formula: see text] a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in [Formula: see text]. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra [Formula: see text], we prove that (C(X) ⋊α ℤ) ⊗ Mq∞ is tracially approximately [Formula: see text] if there exists a y ∈ X such that the C*-subalgebra (C*(C(X), uC0(X\{y}))) ⊗ Mq∞ is tracially approximately [Formula: see text]. If the class [Formula: see text] consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with [Formula: see text] for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either C(X) ⋊α ℤ or C*(C(X), uC0(X\{y})), nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces.

scholarly journals TRACIAL EQUIVALENCE FOR $C^*$-ALGEBRAS AND ORBIT EQUIVALENCE FOR MINIMAL DYNAMICAL SYSTEMS

2005 ◽  
Vol 48 (3) ◽  
pp. 673-690
Author(s):  
Huaxin Lin
Keyword(s):  
Dynamical Systems ◽  
Crossed Products ◽  
Rank Zero ◽  
C Algebras ◽  
Orbit Equivalent ◽  
Linear Maps ◽  
Positive Linear Maps ◽  
Tracial Topological Rank ◽  
Minimal Dynamical Systems ◽  
Cantor Minimal Systems

AbstractWe introduce the notion of tracial equivalence for $C^*$-algebras. Let $A$ and $B$ be two unital separable $C^*$-algebras. If they are tracially equivalent, then there are two sequences of asymptotically multiplicative contractive completely positive linear maps $\phi_n:A\to B$ and $\psi_n:B\to A$ with a tracial condition such that $\{\phi_n\circ\psi_n\}$ and $\{\psi_n\circ\phi_n\}$ are tracially approximately inner. Let $A$ and $B$ be two unital separable simple $C^*$-algebras with tracial topological rank zero. It is proved that $A$ and $B$ are tracially equivalent if and only if $A$ and $B$ have order isomorphic ranges of tracial states. For the Cantor minimal systems $(X_1,\sigma_1)$ and $(X_2,\sigma_2)$, using a result of Giordano, Putnam and Skau, we show that two such dynamical systems are (topological) orbit equivalent if and only if the associated crossed products $C(X_1)\times_{\sigma_1}\mathbb{Z}$ and $C(X_2)\times_{\sigma_2}\mathbb{Z}$ are tracially equivalent.

scholarly journals KMS States on Nica-Toeplitz $$\hbox {C}^*$$-algebras

2020 ◽  
Vol 378 (3) ◽  
pp. 1875-1929
Author(s):  
Zahra Afsar ◽  
Nadia S. Larsen ◽  
Sergey Neshveyev
Keyword(s):  
Complete Classification ◽  
Inverse Temperature ◽  
Toeplitz Algebra ◽  
Gauge Invariant ◽  
Kms States ◽  
C Algebras ◽  
System A ◽  
System Of Inequalities ◽  
Tracial States

Abstract Given a quasi-lattice ordered group (G, P) and a compactly aligned product system X of essential $$\hbox {C}^*$$ C ∗ -correspondences over the monoid P, we show that there is a bijection between the gauge-invariant $$\hbox {KMS}_\beta $$ KMS β -states on the Nica-Toeplitz algebra $$\mathcal {NT}(X)$$ NT ( X ) of X with respect to a gauge-type dynamics, on one side, and the tracial states on the coefficient algebra A satisfying a system (in general infinite) of inequalities, on the other. This strengthens and generalizes a number of results in the literature in several directions: we do not make any extra assumptions on P and X, and our result can, in principle, be used to study KMS-states at any finite inverse temperature $$\beta $$ β . Under fairly general additional assumptions we show that there is a critical inverse temperature $$\beta _c$$ β c such that for $$\beta >\beta _c$$ β > β c all $$\hbox {KMS}_\beta $$ KMS β -states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of $$\hbox {KMS}_\beta $$ KMS β -states in terms of tracial states on A, while at $$\beta =\beta _c$$ β = β c we have a phase transition manifesting itself in the appearance of $$\hbox {KMS}_\beta $$ KMS β -states that are not of Gibbs type. In the case of right-angled Artin monoids we show also that our system of inequalities for traces on A can be reduced to a much smaller system, a finite one when the monoid is finitely generated. Most of our results generalize to arbitrary quasi-free dynamics on $$\mathcal {NT}(X)$$ NT ( X ) .

scholarly journals On Rørdam's classification of certain $C^*$-algebras with one non-trivial ideal, II

2007 ◽  
Vol 101 (2) ◽  
pp. 280 ◽  
Author(s):  
Gunnar Restorff ◽  
Efren Ruiz
Keyword(s):  
Classification Theorem ◽  
C Algebras ◽  
Essential Extensions ◽  
Exact Sequences

In this paper we extend the classification results obtained by Rørdam in the paper [16]. We prove a strong classification theorem for the unital essential extensions of Kirchberg algebras, a classification theorem for the non-stable, non-unital essential extensions of Kirchberg algebras, and we characterize the range in both cases. The invariants are cyclic six term exact sequences together with the class of some unit.

scholarly journals The C*-algebras of Morse—Smale flows on two-manifolds

1990 ◽  
Vol 10 (3) ◽  
pp. 565-597 ◽  
Author(s):  
Xiaolu Wang
Keyword(s):  
Dynamical Systems ◽  
Topological Invariants ◽  
C Algebras ◽  
Smale Flows

AbstractWe give a classification of all the C*-algebras of Morse-Smale flows on closed two-manifolds, and determine the relation between the invariants of dynamical systems and the topological invariants of the C*-algebras.

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

On classification of non-unital amenable simple C*-algebras, II

2020 ◽  
Vol 158 ◽  
pp. 103865
Author(s):  
Guihua Gong ◽  
Huaxin Lin
Keyword(s):  
C Algebras
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