Journal of the Mathematical Society of Japan

Indecomposable characters of infinite dimensional groups associated with operator algebras

Takumi ENOMOTO and Masaki IZUMI

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 determine the indecomposable characters of several classes of infinite dimensional groups associated with operator algebras, including the unitary groups of arbitrary unital simple AF algebras and II1 factors.

Article information

Source
J. Math. Soc. Japan Volume 68, Number 3 (2016), 1231-1270.

Dates
First available in Project Euclid: 19 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1468956167

Digital Object Identifier
doi:10.2969/jmsj/06831231

Mathematical Reviews number (MathSciNet)
MR3523546

Zentralblatt MATH identifier
1353.22011

Subjects
Primary: 22E66: Analysis on and representations of infinite-dimensional Lie groups
Secondary: 46L99: None of the above, but in this section

Keywords
characters infinite dimensional groups AF algebras ergodic method

Citation

ENOMOTO, Takumi; IZUMI, Masaki. Indecomposable characters of infinite dimensional groups associated with operator algebras. J. Math. Soc. Japan 68 (2016), no. 3, 1231--1270. doi:10.2969/jmsj/06831231. https://projecteuclid.org/euclid.jmsj/1468956167.


Export citation

References

  • H. Ando and Y. Matsuzawa, On Polish groups of finite type, Publ. Res. Inst. Math. Sci., 48 (2012), 389–408.
  • B. Blackadar, $K$-theory for operator algebras, Second edition, Mathematical Sciences Research Institute Publications, 5, Cambridge University Press, Cambridge, 1998.
  • R. P. Boyer, Representation theory of the Hilbert-Lie group $U(\mathfrak{H})_2$, Duke Math. J., 47 (1980), 325–344.
  • R. P. Boyer, Infinite traces of AF-algebras and characters of $U(\infty)$, J. Operator Theory, 9 (1983), 205–236.
  • R. P. Boyer, Characters and factor representations of the infinite-dimensional classical groups, J. Operator Theory, 28 (1992), 281–307.
  • R. P. Boyer, Characters and factor representations of the unitary group of the CAR-algebra, J. Operator Theory, 30 (1993), 315–328.
  • K. R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, 6, Amer. Math. Soc., Providence, RI, 1996.
  • A. Dudko, Characters on the full group of an ergodic hyperfinite equivalence relation, J. Funct. Anal., 261 (2011), 1401–1414.
  • A. Dudko and K. Medynets, On characters of inductive limits of symmetric groups, J. Funct. Anal., 264 (2013), 1565–1598.
  • G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra, 38 (1976), 29–44.
  • G. A. Elliott and D. E. Evans, The structure of the irrational rotation $C^*$-algebra, Ann. of Math. (2), 138 (1993), 477–501.
  • E. E. Goryachko and F. V. Petrov, Indecomposable characters of the group of rational rearrangements of a segment, J. Math. Sci. (N. Y.), 174 (2011), 7–14.
  • U. Haagerup and S. Thorbjørnsen, Random matrices and $K$-theory for exact $C^*$-algebras, Doc. Math., 4 (1999), 341–450.
  • Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math., 79 (1957), 87–120.
  • T. Hirai and E. Hirai, Positive definite class functions on a topological group and characters of factor representations, J. Math. Kyoto Univ., 45 (2005), 355–379.
  • T. Hirai and E. Hirai, Characters of wreath products of finite groups with the infinite symmetric group, J. Math. Kyoto Univ., 45 (2005), 547–597.
  • T. Hirai, E. Hirai and A. Hora, Limits of characters of wreath products $\mathfrak{S}_n(T)$ of a compact group $T$ with the symmetric groups and characters of $\mathfrak{S}_\infty(T)$, II, From a viewpoint of probability theory, J. Math. Soc. Japan, 60 (2008), 1187–1217.
  • T. Hirai, E. Hirai and A. Hora, Limits of characters of wreath products $\mathfrak{S}_n(T)$ of a compact group $T$ with the symmetric groups and characters of $\mathfrak{S}_\infty(T)$, I, Nagoya Math. J., 193 (2009), 1–93.
  • C. Itzykson and J. B. Zuber, The planar approximation, II, J. Math. Phys., 21 (1980), 411–421.
  • X. Jiang and H. Su, On a simple unital projectionless $C^*$-algebra, Amer. J. Math., 121 (1999), 359–413.
  • S. V. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis, Translated from the Russian manuscript by N. V. Tsilevich, With a foreword by A. Vershik and comments by G. Olshanski, Translations of Mathematical Monographs, 219, Amer. Math. Soc., Providence, RI, 2003.
  • R. C. King, Branching rules for classical Lie groups using tensor and spinor methods, J. Phys. A, 8 (1975), 429–449.
  • A. A. Kirillov, Representations of the infinite-dimensional unitary group, Dokl. Akad. Nauk. SSSR, 212 (1973), 288–290, English translation: Soviet Math. Dokl., 14 (1973), 1355–1358.
  • H. Lin, An introduction to the classification of amenable $C^*$-algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
  • I. G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  • A. Okounkov and G. Olshanski, Asymptotics of Jack polynomials as the number of variables goes to infinity, Internat. Math. Res. Notices, 1998, 641–682.
  • G. I. Ol'šhanskiĭ, Unitary representations of the infinite-dimensional classical groups $U(p,\infty)$, $SO_0(p,\infty)$, $Sp(p,\infty)$, and of the corresponding motion groups, Funktsional. Anal. i Prilozhen., 12 (1978), 32–44, English translation: Functional Anal. Appl., 12 (1978), 185–195 (1979).
  • G. Olshanski, The problem of harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal., 205 (2003), 464–524.
  • G. K. C. Pedersen, $C^*$-algebras and their automorphism groups, London Mathematical Society Monographs, 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979.
  • H.-L. Skudlarek, Die unzerlegbaren Charaktere einiger diskreter Gruppen, Math. Ann., 233 (1976), 213–231.
  • \c S. Strătilă and D. Voiculescu, Representations of AF-algebras and of the group $U(\infty)$, Lecture Notes in Mathematics, 486, Springer-Verlag, Berlin, 1975.
  • E. Thoma, Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z., 85 (1964), 40–61.
  • A. M. Vershik and S. V. Kerov, Characters and factor representations of the infinite symmetric group, Dokl. Akad. Nauk SSSR, 257 (1981), 1037–1040, English translation: Soviet Math. Dokl., 23 (1981), 389–392.
  • A. M. Vershik and S. V. Kerov, Characters and factor-representations of the infinite unitary group, Doklady AN SSSR, 267 (1982), 272–276, English translation: Soviet Math. Dokl., 26 (1982), 570–574.
  • D. Voiculescu, Sur les représentations factorielles finies de $U(\infty)$ et autres groupes semblables, C. R. Acad. Sci. Paris Sér. A, 279 (1974), 945–946.
  • D. Voiculescu, Représentations factorielles de type II$_1$ de $U(\infty)$, J. Math. Pures Appl. (9), 55 (1976), 1–20.
  • D. P. Želobenko, Compact Lie groups and their representations, Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, 40, Amer. Math. Soc., Providence, RI, 1973.
  • S. Zhang, Matricial structure and homotopy type of simple $C^*$-algebras with real rank zero, J. Operator Theory, 26 (1991), 283–312.