C*-algebras associated to shift spaces

TL;DR

This paper introduces C*-algebras linked to shift spaces, demonstrating their invariance under conjugacy and exploring their K-theory, aimed at readers new to operator algebras.

Contribution

It provides foundational definitions and proofs of invariance properties of C*-algebras associated to shift spaces, with an accessible introduction to related operator algebra concepts.

Findings

C*-algebra associated to a shift space is a one-sided conjugacy invariant.

Morita equivalence class of the C*-algebra is a two-sided conjugacy and flow invariant.

Includes an overview of K-theory for these C*-algebras.

Abstract

These are some notes I wrote for the summer school "Symbolic dynamics and homeomorphisms of the Cantor set" at the University of Copenhagen, 23 - 27 June 2008. The notes contain the definition of the C*-algebra associated to a shift space and some basic facts about these. The notes furthermore contain a proof of the fact that the C*-algebra associated to a shift space is a one-sided conjugacy invariant, and a proof of the fact that the Morita equivalence class of the C*-algebra associated to a shift space is a two-sided conjugacy and a flow invariant. The notes also contain a section (without proofs) about the K-theory of C*-algebras associated to shift spaces. The notes are written for people without a background in operator algebra and contains a short appendix about C*-algebras, Morita equivalence and K-theory of C*-algebras.