Representations of Cuntz algebras associated to quasi-stationary Markov measures

DORIN ERVIN DUTKAY; PALLE E. T. JORGENSEN

doi:10.1017/etds.2014.37

Representations of Cuntz algebras associated to quasi-stationary Markov measures

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.37 ◽

2014 ◽

Vol 35 (7) ◽

pp. 2080-2093 ◽

Cited By ~ 5

Author(s):

DORIN ERVIN DUTKAY ◽

PALLE E. T. JORGENSEN

Keyword(s):

Cuntz Algebra ◽

Unitary Equivalence ◽

Cuntz Algebras ◽

Infinite Words ◽

Markov Measures

In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, as well as the corresponding question of equivalence of associated Cuntz algebra${\mathcal{O}}_{N}$-representations. We do this by associating certain monic representations of${\mathcal{O}}_{N}$to quasi-stationary Markov measures and then proving that equivalence for a pair of measures is decided by unitary equivalence of the corresponding pair of representations.

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The group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra

International Journal of Mathematics ◽

10.1142/s0129167x19500575 ◽

2019 ◽

Vol 30 (11) ◽

pp. 1950057 ◽

Cited By ~ 1

Author(s):

M. Izumi ◽

T. Sogabe

Keyword(s):

Automorphism Group ◽

Vector Bundles ◽

Group Structure ◽

Cuntz Algebra ◽

Classifying Space ◽

Cuntz Algebras ◽

K Theory ◽

Continuous Fields

We determine the group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra [Formula: see text] for finite [Formula: see text] in terms of K-theory. We show that there is an example of a space for which the homotopy set is a noncommutative group, and hence, the classifying space of the automorphism group of the Cuntz algebra for finite [Formula: see text] is not an H-space. We also make an improvement of Dadarlat’s classification of continuous fields of the Cuntz algebras in terms of vector bundles.

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ON AUTOMORPHISMS OF GENERALIZED CUNTZ ALGEBRAS

International Journal of Mathematics ◽

10.1142/s0129167x9800021x ◽

1998 ◽

Vol 09 (04) ◽

pp. 493-512 ◽

Cited By ~ 4

Author(s):

YOSHIKAZU KATAYAMA ◽

HIROAKI TAKEHANA

Keyword(s):

Fixed Point ◽

Finite Type ◽

Invertible Operator ◽

Cuntz Algebra ◽

Gauge Action ◽

Cuntz Algebras ◽

C Algebra ◽

Fixed Point Algebra

Let X be a full right Hilbert B-bimodule of finite type and [Formula: see text] be its generalized Cuntz algebra. We give a notion that the C*-algebra B is X-aperiodic. We show that the fixed point algebra ℱX for a gauge action is simple if and only if the C*-algebra B is X-aperiodic. For a invertible operator U on X with some properties, a quasi-free automorphism αU of [Formula: see text] is defined. We give some conditions in order that αU is inner in the case that B is X-aperiodic. We apply them to the automorphism αU on Cuntz–Krieger algebras.

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Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra On

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is009010003jkt092 ◽

2009 ◽

Vol 6 (2) ◽

pp. 339-380 ◽

Cited By ~ 8

Author(s):

A.L. Carey ◽

J. Phillips ◽

A. Rennie

Keyword(s):

Automorphism Group ◽

General Theory ◽

Index Theory ◽

Index Theorem ◽

Cuntz Algebra ◽

Spectral Flow ◽

Gauge Invariant ◽

Gauge Action ◽

Cuntz Algebras ◽

Modular Automorphism

AbstractThis paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on each Cuntz algebra. We introduce a modified K1-group for each Cuntz algebra which has an index pairing with this twisted cocycle. This index pairing for Cuntz algebras has an interpretation in terms of Araki's notion of relative entropy.

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UNIVERSAL ALGEBRA OF SECTORS

International Journal of Algebra and Computation ◽

10.1142/s0218196709005172 ◽

2009 ◽

Vol 19 (03) ◽

pp. 347-371 ◽

Cited By ~ 2

Author(s):

KATSUNORI KAWAMURA

Keyword(s):

Field Theory ◽

Universal Algebra ◽

Equivalence Classes ◽

Cuntz Algebra ◽

Quantum Field ◽

Unitary Equivalence ◽

C Algebra ◽

Branching Laws ◽

Common Unit ◽

Algebraic Formulation

We show that a nontrivial example of universal algebra appears in quantum field theory. For a unital C *-algebra [Formula: see text], a sector is a unitary equivalence class of unital *-endomorphisms of [Formula: see text]. We show that the set [Formula: see text] of all sectors of [Formula: see text] is a universal algebra with an N-ary sum which is not reduced to any binary sum when [Formula: see text] includes the Cuntz algebra [Formula: see text] as a C *-subalgebra with common unit for N ≥ 3. Next we explain that the set [Formula: see text] of all unitary equivalence classes of unital *-representations of [Formula: see text] is a right module of [Formula: see text]. An essential algebraic formulation of branching laws of representations is given by using submodules of [Formula: see text]. As an application, we show that the action of [Formula: see text] on [Formula: see text] distinguishes elements of [Formula: see text].

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STATES ON THE CUNTZ ALGEBRAS AND p-ADIC RANDOM WALKS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788711001212 ◽

2011 ◽

Vol 90 (2) ◽

pp. 197-211 ◽

Cited By ~ 5

Author(s):

P. E. T. JORGENSEN ◽

A. M. PAOLUCCI

Keyword(s):

Random Walks ◽

Adjacency Matrix ◽

Cuntz Algebras ◽

Ongoing Work ◽

Random Walks On Graphs ◽

Markov Measures

AbstractWe study Markov measures and p-adic random walks with the use of states on the Cuntz algebras Op. Via the Gelfand–Naimark–Segal construction, these come from families of representations of Op. We prove that these representations reflect selfsimilarity especially well. In this paper, we consider a Cuntz–Krieger type algebra where the adjacency matrix depends on a parameter q ( q=1 is the case of Cuntz–Krieger algebra). This is an ongoing work generalizing a construction of certain measures associated to random walks on graphs.

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Endomorphisms of type II1-factors and Cuntz algebras

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870003785x ◽

1996 ◽

Vol 60 (3) ◽

pp. 343-354

Author(s):

Masatoshi Enomoto ◽

Yasuo Watatani

Keyword(s):

Cuntz Algebra ◽

Type Ii ◽

Cuntz Algebras ◽

Jones Index ◽

Standard Space

AbstractAny unital *-endomorphism of a type II1-factor is implemented by isometries of a Cuntz algebra outside the factor. If the Jones index of the range of the *-endomorphism is an integer and the algebras act on the standard space, the Jones index must agree with the number of the generators of the Cuntz algebra. We also study (outer) conjugacy of *-endomorphisms using Cuntz algebras.

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