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.

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.

Topological invariants for substitution tilings and their associated $C^\ast$-algebras

1998 ◽  
Vol 18 (3) ◽  
pp. 509-537 ◽  
Author(s):  
JARED E. ANDERSON ◽  
IAN F. PUTNAM
Keyword(s):  
Dynamical Systems ◽  
Zeta Function ◽  
Inverse Limit ◽  
Topological Invariants ◽  
Two Dimensional ◽  
C Algebras ◽  
Underlying Space ◽  
Substitution Tilings ◽  
Penrose Tilings

We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss $C^*$-algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

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.

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

Classification of the Morse – Smale flows on surfaces with a finite moduli of stability number in sense of topological conjugacy

2021 ◽  
Vol 29 (6) ◽  
pp. 835-850
Author(s):  
Vladislav Kruglov ◽  
◽  
Olga Pochinka ◽  
◽  
Keyword(s):  
Finite Number ◽  
Limit Cycle ◽  
Topological Classification ◽  
Topological Conjugacy ◽  
Smooth Flow ◽  
Direction Of Movement ◽  
Smale Flows ◽  
Phase Surface ◽  
Flows On Surfaces

Purpose. The purpose of this study is to consider the class of Morse – Smale flows on surfaces, to characterize its subclass consisting of flows with a finite number of moduli of stability, and to obtain a topological classification of such flows up to topological conjugacy, that is, to find an invariant that shows that there exists a homeomorphism that transfers the trajectories of one flow to the trajectories of another while preserving the direction of movement and the time of movement along the trajectories; for the obtained invariant, to construct a polynomial algorithm for recognizing its isomorphism and to construct the realisation of the invariant by a standard flow on the surface. Methods. Methods for finding moduli of topological conjugacy go back to the classical works of J. Palis, W. di Melo and use smooth flow lianerization in a neighborhood of equilibrium states and limit cycles. For the classification of flows, the traditional methods of dividing the phase surface into regions with the same behavior of trajectories are used, which are a modification of the methods of A. A. Andronov, E. A. Leontovich, and A. G. Mayer. Results. It is shown that a Morse – Smale flow on a surface has a finite number of moduli if and only if it does not have a trajectory going from one limit cycle to another. For a subclass of Morse – Smale flows with a finite number of moduli, a classification is done up to topological conjugacy by means of an equipped graph. Conclusion. The criterion for the finiteness of the number of moduli of Morse – Smale flows on surfaces is obtained. A topological invariant is constructed that describes the topological conjugacy class of a Morse – Smale flow on a surface with a finite number of modules, that is, without trajectories going from one limit cycle to another.

scholarly journals ON THE CLASSIFICATION OF NUCLEAR C*-ALGEBRAS

2002 ◽  
Vol 85 (1) ◽  
pp. 168-210 ◽  
Author(s):  
MARIUS DADARLAT ◽  
SØREN EILERS
Keyword(s):  
Subject Classification ◽  
Unified Approach ◽  
C Algebras ◽  
Starting Point ◽  
K Theory

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras.2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.

scholarly journals On the classification of C*-algebras of real rank zero, III: The infinite case

1998 ◽  
pp. 11-72 ◽  
Author(s):  
Ola Bratteli ◽  
George Elliott ◽  
David Evans ◽  
Akitaka Kishimoto
Keyword(s):  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebras ◽  
Infinite Case

scholarly journals Classification of rough transformations of a circle from a modern point of view

2018 ◽  
Vol 20 (4) ◽  
pp. 408-418
Author(s):  
A. E. Kolobyanina ◽  
E. V. Nozdrinova ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Point Of View ◽  
Topological Classification ◽  
Topological Invariants ◽  
Modern Theory ◽  
Ambient Manifold ◽  
Periodic Data

In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.

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