An étale equivalence relation on a tiling space arising from a two-sided subshift and associated C*-algebras. Issue 3 (3rd July 2021) - Matsumoto, Kengo
Abstract : A λ -graph bisystem L consists of a pair ( L −, L + ) of two labelled Bratteli diagrams, that presents a two-sided subshift Λ L . We will construct a compact totally disconnected metric space consisting of tilings of a two-dimensional half plane from a λ -graph bisystem. The tiling space has a certain AF-equivalence relation written R L with a natural shift homeomorphism σ L coming from the shift homeomorphism σ Λ L on the subshift Λ L . The equivalence relation R L yields an AF-algebra F L with an automorphism ρ L induced by σ L . We will study invariance of the étale equivalence relation R L, the groupoid G L = R L ⋊ σ L Z and the groupoid C ∗ -algebras C ∗ ( R L ), C ∗ ( G L ) under topological conjugacy of the presenting two-sided subshifts.