Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-19T08:50:58.044Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Amenability of Locally Compact Quantum Groups

Published online by Cambridge University Press:  20 November 2018

Piotr M. Sołtan  and
Ami Viselter
Piotr M. Sołtan
Affiliation:
Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Poland e-mail: piotr.soltan@fuw.edu.pl
Ami Viselter
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 e-mail: viselter@ualberta.ca
Rights & Permissions [Opens in a new window]

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.

In this short note we introduce a notion called quantum injectivity of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. In particular, this provides a new characterization of amenability of locally compact groups.

Keywords

20G4222D2546L89amenabilityconditional expectationinjectivitylocally compact quantum groupquantum injectivity
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 2 , 14 June 2014 , pp. 424 - 430
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

The first author was partially supported by the National Science Center (NCN) grant no. 2011/01/B/ST1/05011. The second author was supported by NSERC Discovery Grant no. 418143-2012.

References

[1] Bédos, E., Murphy, G. J., and Tuset, L., Amenability and coamenability of algebraic quantum groups. Int. J. Math. Math. Sci. 31 (2002, no. 10, 577–601.http://dx.doi.org/10.1155/S016117120210603X CrossRefGoogle Scholar
[2] Bédos, E. and Tuset, L., Amenability and co-amenability for locally compact quantum groups. Internat. J. Math. 14 (2003, no. 8, 865–884.http://dx.doi.org/10.1142/S0129167X03002046 CrossRefGoogle Scholar
[3] Connes, A., Classification of injective factors. Cases II 1; II1; III; 6= 1. Ann. of Math. (2) 104 (1976, no. 1, 73–115.http://dx.doi.org/10.2307/1971057 Google Scholar
[4] Crann, J. and Neufang, M., Quantum group amenability, injectivity, and a question of Bédos–Tuset. arxiv:1208.2986Google Scholar
[5] Desmedt, P., Quaegebeur, J., and Vaes, S., Amenability and the bicrossed product construction. Illinois J. Math. 46 (2002, no. 4, 1259–1277.Google Scholar
[6] Doplicher, S., Longo, R., Roberts, J. E., and Zsidö, L., A remark on quantum group actions and nuclearity. Rev. Math. Phys. 14 (2002, no. 7-8, 787–796.http://dx.doi.org/10.1142/S0129055X02001399 Google Scholar
[7] Effros, E. G. and Kishimoto, A., Module maps and Hochschild-Johnson cohomology. Indiana Univ. Math. J. 36 (1987, no. 2, 257–276. http://dx.doi.org/10.1512/iumj.1987.36.36015 Google Scholar
[8] Enock, M. and Schwartz, J.-M., Algèbres de Kac moyennables. Pacific J. Math. 125 (1986, no. 2, 363–379. http://projecteuclid.org/euclid.pjm/1102700082Google Scholar
[9] Fima, P., On locally compact quantum groups whose algebras are factors. J. Funct. Anal. 244 (2007, no. 1, 78–94.http://dx.doi.org/10.1016/j.jfa.2006.03.002 Google Scholar
[10] Kustermans, J. and Vaes, S., Locally compact quantum groups. Ann. Sci. E´ cole Norm. Sup. (4) 33 (2000, no. 6, 837–934.http://dx.doi.org/10.1016/S0012-9593(00)01055-7 Google Scholar
[11] Kustermans, J. and Vaes, S., Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92 (2003, no. 1, 68–92.Google Scholar
[12] Paterson, A. L. T., Amenability. Mathematical Surveys and Monographs, 29, American Mathematical Society, Providence, RI, 1988.Google Scholar
[13] Ruan, Z.-J., Amenability of Hopf von Neumann algebras and Kac algebras. J. Funct. Anal. 139 (1996, no. 2, 466–499.http://dx.doi.org/10.1006/jfan.1996.0093 Google Scholar
[14] Runde, V., Lectures on amenability. Lecture Notes in Mathematics, 1774, Springer-Verlag, Berlin, 2002.Google Scholar
[15] Runde, V., Characterizations of compact and discrete quantum groups through second duals. J. Operator Theory 60 (2008, no. 2, 415–428.Google Scholar
[16] Sołtan, P. M., Quantum Bohr compactification. Illinois J. Math. 49 (2005, no. 4, 1245–1270.Google Scholar
[17] Takesaki, M., Theory of operator algebras. I. Encyclopaedia of Mathematical Sciences, 124, Operator Algebras and Non-commutative Geometry, 5, Springer-Verlag, Berlin, 2002. [18] R. Tomatsu, Amenable discrete quantum groups. J. Math. Soc. Japan 58 (2006, no. 4, 949–964.http://dx.doi.org/10.2969/jmsj/1179759531 Google Scholar
[19] Woronowicz, S. L., Compact quantum groups. In: Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 845–884.Google Scholar