On Certain K-Groups Associated with Minimal Flows

1998 ◽  
Vol 41 (2) ◽  
pp. 240-244
Author(s):  
Jingbo Xia
Keyword(s):  
Toeplitz Operators ◽  
General Setting ◽  
Unique Ergodicity ◽  
Toeplitz Algebra ◽  
Commutator Ideal ◽  
Uniquely Ergodic

AbstractIt is known that the Toeplitz algebra associated with any flow which is both minimal and uniquely ergodic always has a trivial K1-group. We show in this note that if the unique ergodicity is dropped, then such K1-group can be non-trivial. Therefore, in the general setting of minimal flows, even the K-theoretical index is not sufficient for the classification of Toeplitz operators which are invertible modulo the commutator ideal.

scholarly journals BERGMAN KERNELS AND EQUILIBRIUM MEASURES FOR POLARIZED PSEUDO-CONCAVE DOMAINS

2010 ◽  
Vol 21 (01) ◽  
pp. 77-115 ◽  
Author(s):  
ROBERT J. BERMAN
Keyword(s):  
Toeplitz Operators ◽  
Equilibrium Measure ◽  
Markov Property ◽  
General Setting ◽  
Space Structure ◽  
Tensor Power ◽  
Bergman Kernels ◽  
Equilibrium Measures ◽  
Holomorphic Sections ◽  
Positive Line Bundle

Let X be a domain in a closed polarized complex manifold (Y,L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form ωn on X. In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X,ωn) is replaced by any measure satisfying a Bernstein–Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.

Stationary processes and pure point diffraction

2016 ◽  
Vol 37 (8) ◽  
pp. 2597-2642 ◽  
Author(s):  
DANIEL LENZ ◽  
ROBERT V. MOODY
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
General Setting ◽  
Pure Point ◽  
Stationary Processes ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Point Measure

We consider the construction and classification of some new mathematical objects, called ergodic spatial stationary processes, on locally compact abelian groups. These objects provide a natural and very general setting for studying diffraction and the famous inverse problems associated with it. In particular, we can construct complete families of solutions to the inverse problem from any given positive pure point measure that is chosen to be the diffraction. In this case these processes can be classified by the dual of the group of relators based on the set of Bragg peaks, and this gives an abstract solution to the homometry problem for pure point diffraction.

scholarly journals Amenability and unique ergodicity of automorphism groups of countable homogeneous directed graphs

2018 ◽  
Vol 40 (5) ◽  
pp. 1351-1401
Author(s):  
MICHEAL PAWLIUK ◽  
MIODRAG SOKIĆ
Keyword(s):  
Automorphism Groups ◽  
Directed Graphs ◽  
Point Of View ◽  
Unique Ergodicity ◽  
Minimal Flow ◽  
Negative Results ◽  
Starting Point ◽  
Complete Multipartite ◽  
Uniquely Ergodic

We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable (in their natural topologies). For those which are amenable, we determine whether they are uniquely ergodic, leaving unsettled precisely one case (the ‘semi-generic’ complete multipartite directed graph). We also consider the Hrushovski property. For most of our results we use the various techniques of Angelet al[Random orderings and unique ergodicity of automorphism groups.J. Eur. Math. Soc.,16(2014), 2059–2095], suitably generalized to a context in which the universal minimal flow is not necessarily the space of all orders. Negative results concerning amenability rely on constructions of the type considered in Zucker [Amenability and unique ergodicity of automorphism groups of Fraïssé structures.Fund. Math.,226(2014), 41–61]. An additional class of structures (compositions) may be handled directly on the basis of very general principles. The starting point in all cases is the determination of the universal minimal flow for the automorphism group, which in the context of countable homogeneous directed graphs is given in Jasińskiet al[Ramsey precompact expansions of homogeneous directed graphs.Electron. J. Combin.,21(4), (2014), 31] and the papers cited therein.

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 Unique ergodicity for non-uniquely ergodic horocycle flows

2006 ◽  
Vol 16 (2) ◽  
pp. 411-433 ◽  
Author(s):  
François Ledrappier ◽  
◽  
Omri Sarig ◽  
Keyword(s):  
Unique Ergodicity ◽  
Uniquely Ergodic

scholarly journals Boshernitzan's criterion for unique ergodicity of an interval exchange transformation

1987 ◽  
Vol 7 (1) ◽  
pp. 149-153 ◽  
Author(s):  
William A. Veech
Keyword(s):  
Unique Ergodicity ◽  
Interval Exchange ◽  
Interval Exchange Transformation ◽  
Exchange Transformation ◽  
Uniquely Ergodic

AbstractConfirming a conjecture by Boshernitzan, it is proved that ifTis a minimal non-uniquely ergodic interval exchange, the minimum spacing of the partition determined byTnis O(1/n).

scholarly journals Unique ergodicity of the automorphism group of the semigeneric directed graph

2021 ◽  
pp. 1-14
Author(s):  
COLIN JAHEL
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Unique Ergodicity ◽  
Uniquely Ergodic

Abstract We prove that the automorphism group of the semigeneric directed graph (in the sense of Cherlin’s classification) is uniquely ergodic.

A context in which finite or unique ergodicity is generic

2020 ◽  
pp. 1-23
Author(s):  
ANDY Q. YINGST
Keyword(s):  
Unique Ergodicity ◽  
Cantor Space ◽  
Bernoulli Trial ◽  
Residual Set ◽  
Ergodic Measures ◽  
Uniquely Ergodic

Abstract We show that for good measures, the set of homeomorphisms of Cantor space which preserve that measure and which have no invariant clopen sets contains a residual set of homeomorphisms which are uniquely ergodic. Additionally, we show that for refinable Bernoulli trial measures, the same set of homeomorphisms contains a residual set of homeomorphisms which admit only finitely many ergodic measures.

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 ) .

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