Subshifts, λ-graph bisystems and C⁎-algebras

Abstract

We introduce a notion of λ-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 λ-graph system, that has been previously introduced by the author. Its matrix presentation is called a symbolic matrix bisystem. We first show that any λ-graph bisystem presents subshifts and conversely any subshift is presented by a λ-graph bisystem, called the canonical λ-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 λ-graph bisystem (L−,L+) yields a pair of C⁎-algebras written OL−+,OL+− that are first defined as the C⁎-algebras of certain étale 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 λ-graph bisystem (L−,L+). If a λ-graph bisystem comes from a λ-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift (ΛA,σA) by its transition matrix A of the graph, and the associated C⁎-algebra OL−+ is isomorphic to the Cuntz–Krieger algebra OA, whereas the other C⁎-algebra OL+− is isomorphic to the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z of the commutative C⁎-algebra C(ΛA) of continuous functions on the shift space ΛA of the two-sided topological Markov shift by the automorphism σA⁎ induced by the homeomorphism of the shift σA. This phenomenon shows a duality between Cuntz–Krieger algebra OA and the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z.