Monic representations of the Cuntz algebra and Markov measures

    Authors

    D. E. Dutkay;P. E. T. Jorgensen

    Comments

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    Abbreviated Journal Title

    J. Funct. Anal.

    Keywords

    Wavelet representation; Infinite product measures; Markov measures; Cuntz algebras; ITERATED FUNCTION SYSTEMS; BRATTELI DIAGRAMS; CSTAR-ALGEBRAS; FRACTALS; OPERATORS; WAVELETS; Mathematics

    Abstract

    We study representations of the Cuntz algebras O-N While, for fixed N, the set of equivalence classes of representations of O-N is known not to have a Borel cross section, there are various subclasses of representations which can be classified. We study monic representations of O-N, that have a cyclic vector for the canonical abelian subalgebra. We show that O-N has a certain universal representation which contains all positive monic representations. A large class of examples of monic representations is based on Markov measures. We classify them and as a consequence we obtain that different parameters yield mutually singular Markov measure, extending the classical result of Kakutani. The monic representations based on the Kakutani measures are exactly the ones that have a one-dimensional cyclic S-i*-invariant space. (C) 2014 Elsevier Inc. All rights reserved.

    Journal Title

    Journal of Functional Analysis

    Volume

    267

    Issue/Number

    4

    Publication Date

    1-1-2014

    Document Type

    Article

    Language

    English

    First Page

    1011

    Last Page

    1034

    WOS Identifier

    WOS:000337934200004

    ISSN

    0022-1236

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