Subshifts, λ-graph bisystems and C*-algebras

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:485

Article

Matsumoto, Kengo¹ 

[1] Joetsu Univ Educ, Dept Math, Joetsu 9438512, Japan

关键词: Subshift;    Bratteli diagram;    Etale groupoid;    C*-algebra;    Cuntz-Krieger algebra;   

DOI  :  10.1016/j.jmaa.2020.123843

来源: Elsevier

【 摘 要 】

We introduce a notion of lambda-graph bisystem that consists of a pair (L-, L+) of two labeled Bratteli diagrams L-, L+ satisfying certain compatibility condition for labeling their edges. It is a two-sided extension of lambda-graph system, that has been previously introduced by the author. Its matrix presentation is called a symbolic matrix bisystem. We first show that any lambda-graph bisystem presents subshifts and conversely any subshift is presented by a lambda-graph bisystem, called the canonical lambda-graph bisystem for the subshift. We introduce an algebraically defined relation on symbolic matrix bisystems called properly strong shift equivalence and show that two subshifts are topologically conjugate if and only if their canonical symbolic matrix bisystems are properly strong shift equivalent. A lambda-graph bisystem (L-, L+) yields a pair of C*-algebras written (O-L-(+), O-L+(-)) that are first defined as the C*-algebras of certain etale groupoids constructed from (L-, L+). We study structure of the C*-algebras, and show that they are universal unital unique C*-algebras subject to certain operator relations among canonical generators of partial isometries and projections encoded by the structure of the lambda-graph bisystem (L-, L+). If a lambda-graph bisystem comes from a lambda-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift (Lambda(A), sigma(A)) by its transition matrix A of the graph, and the associated C*-algebra O-L-(+) is isomorphic to the Cuntz-Krieger algebra O-A, whereas the other C*-algebra O-L+(-) is isomorphic to the crossed product C*-algebra C(Lambda(A)) (sic)(sigma*A) Z of the commutative C*-algebra C(Lambda(A)) of continuous functions on the shift space Lambda(A) of the two-sided topological Markov shift by the automorphism sigma(A)* induced by the homeomorphism of the shift sigma(A). This phenomenon shows a duality between Cuntz-Krieger algebra O-A and the crossed product C*-algebra C(Lambda(A)) (sic)(sigma*A) Z. (C) 2020 Elsevier Inc. All rights reserved.

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