Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-08T19:32:12.735Z Has data issue: false hasContentIssue false
  • English
  • Français

A NONSEPARABLE AMENABLE OPERATOR ALGEBRA WHICH IS NOT ISOMORPHIC TO A $C^*$-ALGEBRA

Part of: Set theory Selfadjoint operator algebras Linear spaces and algebras of operators

Published online by Cambridge University Press:  10 March 2014

YEMON CHOI ,
ILIJAS FARAH  and
NARUTAKA OZAWA
YEMON CHOI
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N [email protected]
ILIJAS FARAH
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J [email protected] Matematicki Institut, Kneza Mihaila 35, Belgrade, Serbia
NARUTAKA OZAWA
Affiliation:
RIMS, Kyoto University, 606-8502 Japan; [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It has been a long-standing question whether every amenable operator algebra is isomorphic to a (necessarily nuclear) $\mathrm{C}^*$-algebra. In this note, we give a nonseparable counterexample. Finding out whether a separable counterexample exists remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in $\mathrm{C}^*$-algebras and show that our method cannot produce a separable counterexample.

MSC classification

Secondary: 47L30: Abstract operator algebras on Hilbert spaces 46L05: General theory of $C^*$-algebras 03E75: Applications of set theory
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e2
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

References

Blecher, D. P. and Le Merdy, C. 2004 Operator Algebras and Their Modules—an Operator Space Approach. (London Mathematical Society Monographs. New Series, 30) , Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford.Google Scholar
Choi, Y. ‘On commutative, operator amenable subalgebras of finite von Neumann algebras’, J. Reine Angew. Math. 678, (2013), 201–222.Google Scholar
Connes, A. ‘On the cohomology of operator algebras’, J. Funct. Anal. 28, (1978), 248–253.Google Scholar
Curtis, P. C. Jr and Loy, R. J. ‘A note on amenable algebras of operators’, Bull. Austral. Math. Soc. 52, (1995), 327–329.Google Scholar
Farah, I. and Hart, B. ‘Countable saturation of corona algebras’, C. R. Math. Rep. Acad. Sci. Canada 35, (2013), 35–56.Google Scholar
Gifford, J. A. ‘Operator algebras with a reduction property’, J. Aust. Math. Soc. 80, (2006), 297–315.CrossRefGoogle Scholar
Haagerup, U. ‘All nuclear $\mathrm{C}^*$ -algebras are amenable’, Invent. Math. 74, (1983), 305–319.Google Scholar
Johnson, B. E. 1972 In Cohomology in Banach Algebras Memoirs of the American Mathematical Society, 127, American Mathematical Society, Providence, RI.Google Scholar
Kirchberg, E. ‘On subalgebras of the CAR-algebra’, J. Funct. Anal. 129, (1995), 35–63.Google Scholar
Luzin, N. ‘O chastyah naturalp1nogo ryada’, Izv. AN SSSR, seriya mat. 11 (5) (1947), 714–722 Available at http://www.mathnet.ru/links/55625359125306fbfba1a9a6f07523a0/im3005.pdf.Google Scholar
Monod, N. 2001 Continuous Bounded Cohomology of Locally Compact Groups. vol. 1758. Springer.CrossRefGoogle Scholar
Monod, N. and Ozawa, N. ‘The Dixmier problem, lamplighters and Burnside groups’, J. Funct. Anal. 258, (2010), 255–259.Google Scholar
Ormes, N. S. ‘Real coboundaries for minimal Cantor systems’, Pacific J. Math. 195, (2000), 453–476.Google Scholar
Pedersen, G. K. 1990 ‘The corona construction’, In Operator Theory: Proceedings of the 1988 GPOTS-Wabash Conference, Indianapolis, IN, 1988 Pitman Res. Notes Math. Ser., vol. 225, pp. 49–92. Longman Sci. Tech, Harlow.Google Scholar
Pisier, G. 2001 ‘Similarity problems and completely bounded maps’, In Includes the Solution to The Halmos Problem, Second, expanded edition Lecture Notes in Mathematics, vol. 1618,. Springer-Verlag, Berlin.Google Scholar
Pisier, G. ‘Simultaneous similarity, bounded generation and amenability’, Tohoku Math. J. 59 (2) (2007), 79–99.Google Scholar
Runde, V. 2002 In Lectures on Amenability Lecture Notes in Mathematics, vol. 1774, Springer-Verlag, Berlin.Google Scholar
Šeĭnberg, M. V. ‘A characterization of the algebra $C(\Omega )$ in terms of cohomology groups’, Uspekhi Mat. Nauk 32, (1977), 203–204.Google Scholar
Talayco, D. E. ‘Applications of cohomology to set theory I: Hausdorff gaps’, Ann. Pure Appl. Logic 71, (1995), 69–106.Google Scholar
Voiculescu, D. ‘A note on quasi-diagonal $\mathrm{C}^*$ -algebras and homotopy’, Duke Math. J. 62, (1991), 267–271.Google Scholar
Willis, G. A. ‘When the algebra generated by an operator is amenable’, J. Operator Theory 34, (1995), 239–249.Google Scholar