Annals of Functional Analysis

On graph algebras from interval maps

C. Correia Ramos, Nuno Martins, and Paulo R. Pinto

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We produce and study a family of representations of relative graph algebras on Hilbert spaces that arise from the orbits of points of 1-dimensional dynamical systems, where the underlying Markov interval maps f have escape sets. We identify when such representations are faithful in terms of the transitions to the escape subintervals.

Article information

Source
Ann. Funct. Anal., Volume 10, Number 2 (2019), 203-217.

Dates
Received: 16 March 2018
Accepted: 18 July 2018
First available in Project Euclid: 19 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.afa/1552960867

Digital Object Identifier
doi:10.1215/20088752-2018-0019

Mathematical Reviews number (MathSciNet)
MR3941382

Subjects
Primary: 46L05: General theory of $C^*$-algebras
Secondary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx]

Keywords
graph C$^{\ast}$-algebra representation transition matrix interval map

Citation

Ramos, C. Correia; Martins, Nuno; Pinto, Paulo R. On graph algebras from interval maps. Ann. Funct. Anal. 10 (2019), no. 2, 203--217. doi:10.1215/20088752-2018-0019. https://projecteuclid.org/euclid.afa/1552960867


Export citation

References

  • [1] M. Abe and K. Kawamura, Recursive fermion system in Cuntz algebra, I: Embeddings of fermion algebra into Cuntz algebra, Comm. Math. Phys. 228 (2002), no. 1, 85–101.
  • [2] L. Bandeira and C. Correia Ramos, Transition matrices characterizing a certain totally discontinuous map of the interval, J. Math. Anal. Appl. 444 (2016), no. 2, 1274–1303.
  • [3] T. Bates, D. Pask, I. Raeburn, and W. Szymański, The C$^{\ast}$-algebras of row-finite graphs, New York J. Math. 6 (2000), 307–324.
  • [4] 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.
  • [5] 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.
  • [6] C. Correia Ramos, N. Martins, and P. R. Pinto, Toeplitz algebras arising from escape points of interval maps, Banach J. Math. Anal. 11 (2017), no. 3, 536–553.
  • [7] C. Correia Ramos, N. Martins, and P. R. Pinto, Escape dynamics for interval maps, preprint, 2017.
  • [8] 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.
  • [9] J. Cuntz and W. Krieger, A class of $C^{*}$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268.
  • [10] D. E. Dutkay and P. E. T. Jorgensen, Wavelet constructions in non-linear dynamics, Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 21–33.
  • [11] K. Falconer, Techniques in Fractal Geometry, Wiley, Chichester, 1997.
  • [12] C. Farsi, E. Gillaspy, S. Kang, and J. A. Packer, Separable representations, KMS states, and wavelets for higher-rank graphs, J. Math. Anal. Appl. 434 (2016), no. 1, 241–270.
  • [13] N. J. Fowler and I. Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), no. 1, 155–181.
  • [14] D. Gonçalves and D. Royer, Unitary equivalence of representations of algebras associated with graphs, and branching systems (in Russian), Funktsional. Anal. i Prilozhen. 45 (2011), no. 2, 45–59; English translation in Funct. Anal. Appl. 45, (2011), no. 2, 117–127.
  • [15] P. E. T. Jorgensen, “Certain representations of the Cuntz relations, and a question on wavelets decompositions” in Operator Theory, Operator Algebras, and Applications, Contemp. Math. 414, Amer. Math. Soc., Providence, 2006, 165–188.
  • [16] M. H. Mann, I. Raeburn, and C. E. Sutherland, Representations of finite groups and Cuntz–Krieger algebras, Bull. Aust. Math. Soc. 46 (1992), no. 2, 225–243.
  • [17] M. Marcolli and A. M. Paolucci, Cuntz–Krieger algebras and wavelets on fractals, Complex Anal. Oper. Theory 5 (2011), no. 1, 41–81.
  • [18] P. S. Muhly and M. Tomforde, Adding tails to $C^{*}$-correspondences, Doc. Math. 9 (2004), 79–106.
  • [19] I. Raeburn, Graph Algebras, CBMS Reg. Conf. Ser. Math. 103, Amer. Math. Soc., Providence, 2005.