Ergodic actions of convergent Fuchsian groups on quotients of the noncommutative Hardy algebras

Arias, Alvaro; Latrémolière, Frédéric

doi:10.1090/S0002-9939-2011-10794-9

Ergodic actions of convergent Fuchsian groups on quotients of the noncommutative Hardy algebras

Authors: Alvaro Arias and Frédéric Latrémolière
Journal: Proc. Amer. Math. Soc. 139 (2011), 2485-2496
MSC (2010): Primary 47L15, 47L55; Secondary 32M05
DOI: https://doi.org/10.1090/S0002-9939-2011-10794-9
Published electronically: January 4, 2011
MathSciNet review: 2784814
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish that particular quotients of the noncommutative Hardy algebras carry ergodic actions of convergent discrete subgroups of the group $\operatorname*{SU}(n,1)$ of automorphisms of the unit ball in $\mathbb{C}^{n}$ . To do so, we provide a mean to compute the spectra of quotients of noncommutative Hardy algebra and characterize their automorphisms in terms of biholomorphic maps of the unit ball in $\mathbb{C}^{n}.$

1. A. Arias and F. Latrémoliere, Isomorphisms of non-commutative domain algebras, to appear in J. Oper. Theory.
2. W. Arveson, Subalgebras of -algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159-228. MR 1668582 (2000e:47013)
3. K. Davidson and D. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275-303. MR 1625750 (2001c:47082)
4. -, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc. (3) 78 (1999), no. 2, 401-430. MR 1665248 (2000k:47005)
5. S. Krantz, Function theory of several complex variables, Mathematics Series, Wadsworth and Brooks/Cole, 1992.
6. -, Handbook of complex variables, Birkhäuser, Boston, MA, 1999. MR 1738432 (2001a:30001)
7. G. Popescu, Von Neumann inequality for $(B(\mathcal{H})^n)_1$ , Math. Scand. 68 (1991), no. 2, 292-304. MR 1129595 (92k:47073)
8. -, Non-commutative disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2137-2148. MR 1343719 (96k:47077)
9. -, Poisson transforms on some -algebras generated by isometries, J. Funct. Anal. 161 (1999), no. 1, 27-61. MR 1670202 (2000m:46117)
10. -, Operator theory on noncommutative domains, Mem. Amer. Math. Soc. 205 (2010), no. 964, vi + 124 pp.
11. -, Free holomorphic automorphisms of the unit ball of $B(\mathcal{H})^n$ , J. Reine Angew. Math. 638 (2010), 119-168. MR 2595338
12. W. Rudin, Function theory in the unit ball of $\mathbf {C}^n$ , Grundlehren der Mathematischen Wissenschaften, vol. 241, Springer-Verlag, 1980. MR 0601594 (82i:32002)
13. D. Voiculescu, Symmetries of some reduced free product $C^\ast$ -algebras, Lecture Notes in Mathematics, vol. 1132, Springer-Verlag, Berlin, 1985. MR 799593 (87d:46075)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47L15, 47L55, 32M05

Retrieve articles in all journals with MSC (2010): 47L15, 47L55, 32M05

Additional Information

Alvaro Arias
Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
Email: aarias@math.du.edu

Frédéric Latrémolière
Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
Email: frederic@math.du.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-10794-9
Keywords: Noncommutative Hardy algebras, Fock spaces, Fuchsian group, Mobius transformations, automorphisms of operator algebras.
Received by editor(s): March 2, 2010
Received by editor(s) in revised form: June 30, 2010
Published electronically: January 4, 2011
Communicated by: Marius Junge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.