C^* -ALGEBRAS ASSOCIATED WITH TWO-SIDED SUBSHIFTS. (18th April 2022) - MATSUMOTO, KENGO
Abstract: This paper is a continuation of the paper, Matsumoto ['Subshifts, λ -graph bisystems and C^* -algebras', J. Math. Anal. Appl. 485 (2020), 123843]. A λ -graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift Λ , there exists a λ -graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra F_L with shift automorphism ρ _L from a λ -graph bisystem (L^-, L^+), and define a C^* -algebra R_L by the crossed product . It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If λ -graph bisystems come from two-sided subshifts, these C^* -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the C^* -algebra R_L and the K-theory formulas of the C^* -algebras F_L and R_L . The K-group for the AF-algebra F_L is regarded as a two-sided extension of the dimension group of subshifts.