## Banach Journal of Mathematical Analysis

### Toeplitz algebras arising from escape points of interval maps

#### Abstract

We generate a representation of the Toeplitz $C^{\ast}$-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.

#### Article information

Source
Banach J. Math. Anal., Volume 11, Number 3 (2017), 536-553.

Dates
Accepted: 23 August 2016
First available in Project Euclid: 29 April 2017

https://projecteuclid.org/euclid.bjma/1493431219

Digital Object Identifier
doi:10.1215/17358787-2017-0005

Mathematical Reviews number (MathSciNet)
MR3679895

Zentralblatt MATH identifier
1380.46049

#### Citation

Correia Ramos, C.; Martins, Nuno; Pinto, Paulo R. Toeplitz algebras arising from escape points of interval maps. Banach J. Math. Anal. 11 (2017), no. 3, 536--553. doi:10.1215/17358787-2017-0005. https://projecteuclid.org/euclid.bjma/1493431219

#### References

• [1] A. an Huef and I. Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 611–624.
• [2] O. Bratteli and P. E. T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (1999), no. 663.
• [3] C. Correia Ramos, N. Martins, and P. R. Pinto, Interval maps from Cuntz-Krieger algebras, J. Math. Anal. Appl. 374 (2011), no. 2, 347–354.
• [4] C. Correia Ramos, N. Martins, P. R. Pinto, and J. Sousa Ramos, Cuntz-Krieger algebras representations from orbits of interval maps, J. Math. Anal. Appl. 341 (2008), no. 2, 825–833.
• [5] J. Cuntz and W. Krieger, A class of $C^{\ast}$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268.
• [6] D. E. Evans, Gauge actions on $\mathcal{O}_{A}$, J. Operator Theory 7 (1982), no. 1, 79–100.
• [7] D. E. Evans, The $C^{\ast}$-algebra of topological Markov chains, lectures at Tokyo Metropolitan University, Tokyo, Japan, 1983.
• [8] R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999), 119–172.
• [9] K. Falconer, Techniques in Fractal Geometry, Wiley, Chichester, 1997.
• [10] N. J. Fowler and I. Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), no. 1, 155–181.
• [11] P. S. Muhly and M. Tomforde, Adding tails to $C^{\ast}$-correspondences, Doc. Math. 9 (2004), 79–106.
• [12] G. K. Pedersen, $C^{\ast}$-algebras and Their Automorphism Groups, London Math. Soc. Monogr. 14, Academic Press, London, 1979.
• [13] A. Sims and S. B. G. Webster, A direct approach to co-universal algebras associated to directed graphs, Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 2, 211–220.