Banach Journal of Mathematical Analysis

Baumslag-Solitar group C*-algebras from interval maps

C. Correia Ramos, R. El Harti, 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 yield operators $U$ and $V$ on Hilbert spaces that are parameterized by the orbits of certain interval maps that exhibit chaotic behavior and obey the (deformed) Baumslag--Solitar relation $$UV=e^{2\pi i \alpha} VU^n,\qquad \alpha\in \mathbb{R},\ n\in\mathbb{N}.$$ We then prove that the scalar $e^{2\pi i \alpha}$ can be removed whilst retaining the isomorphism class of the $C^*$-algebra generated by $U$ and $V$. Finally, we simultaneously unitarize $U$ and $V$ by gluing pairs of orbits of the underlying noninvertible dynamical system and investigate these unitary representations under distinct pairs of orbits.

Article information

Source
Banach J. Math. Anal., Volume 8, Number 1 (2014), 138-147.

Dates
First available in Project Euclid: 14 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1381782093

Digital Object Identifier
doi:10.15352/bjma/1381782093

Mathematical Reviews number (MathSciNet)
MR3161688

Zentralblatt MATH identifier
1296.46056

Subjects
Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Secondary: 46L05: General theory of $C^*$-algebras 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx] 37A20: Orbit equivalence, cocycles, ergodic equivalence relations

Keywords
group $C^*$-algebras representations of $C^*$-algebras symbolic dynamics interval maps

Citation

Correia Ramos, C.; El Harti, R.; Martins, Nuno; Pinto, Paulo R. Baumslag-Solitar group C*-algebras from interval maps. Banach J. Math. Anal. 8 (2014), no. 1, 138--147. doi:10.15352/bjma/1381782093. https://projecteuclid.org/euclid.bjma/1381782093


Export citation

References

  • G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199–201.
  • O. Bratteli and P.E.T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 663 (1999), 1–89.
  • O. Bratteli, P.E.T. Jorgensen and V. Ostrovskyi, Representation theory and numerical AF-invariants. The representations and centralizers of certain states on $\mathcal{O}_d$, Mem. Amer. Math. Soc. 168, no. 797, xviii+178 pp., 2004.
  • D.E. Dutkay, Low-pass filters and representations of the Baumslag Solitar group, Trans. Amer. Math. Soc. 358 (2006), 5271–5291.
  • D.E. Dutkay and P.E.T. Jorgensen, A duality approach to representations of Baumslag-Solitar groups, Group representations, ergodic theory, and mathematical physics: a tribute to George W. Mackey, 99–127, Contemp. Math., 449, Amer. Math. Soc., Providence, RI, 2008. Preprint http://arxiv.org/abs/0704.2050v3.
  • 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), 825–833.
  • C. Correia Ramos, N. Martins and P.R. Pinto, Orbit representations and circle maps, In Birkhauser book series on Oper. Algebras, Oper. Theory and Applications Vol. 181 (2008), 417 – 427.
  • C. Correia Ramos, N. Martins and P.R. Pinto, On $C^*$-algebras from interval maps, Complex Anal. Oper. Theory 7 (2013), 221–235.
  • R. El Harti and P.R. Pinto, Stability results for $C^*$-unitarizable groups, Ann. Funct. Anal. \textbf 2 (2011), 1–10.
  • N. Martins and J. Sousa Ramos, Cuntz-Krieger algebras arising from linear mod one transformations, Fields Inst. Commun. 31 (2002), 265–273.
  • D. Moldavanskii, On the isomorphisms of Baumslag-Solitar groups, (Russian Ukrainian summary) Ukrain. Mat. Zh. 43 (1991) 1684–1686; translation in Ukrainian Math. J. 43 (1992), 1569–1571.
  • J. Milnor and W. Thurston, On Iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87), Lect. Notes in Math, 1342 (1988), 465–563.
  • G.K. Pedersen, $C^*$-Algebras and their Automorphism Groups, Academic Press, London Mathematical Society Monographs, 14, ix+416, 1979.
  • M.A. Rieffel, $C^\ast$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415–429.