Toeplitz algebras arising from escape points of interval maps

Abstract

We generate a representation of the Toeplitz $C^{*}$-algebra ${\mathcal{T}}_{A_f}$ on a Hilbert space $H_x$ that encodes the orbit of an escape point $x\in I$ of a Markov interval map $f$, with transition matrix $A_f$. This leads to a family of representations of ${\mathcal{T}}_{A_f}$ labeled by points in all intervals $I$. The underlying dynamics of the interval map are used in the study of this family.