Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T04:55:21.814Z Has data issue: false hasContentIssue false

COMPACT ELEMENTS AND OPERATORS OF QUANTUM GROUPS

Published online by Cambridge University Press:  10 June 2016

MASSOUD AMINI
Affiliation:
Department of Mathematics, Tarbiat Modares University, P.O.Box 14115-134, Tehran, Iran e-mail: [email protected]
MEHRDAD KALANTAR
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, ul. Sniadeckich 8, 00–956 Warszawa, Poland
ALIREZA MEDGHALCHI
Affiliation:
Department of Mathematics, Kharazmi University (Tarbiat Moallem University), 50, Taleghani Ave., 15618, Tehran, Iran e-mail: [email protected], [email protected]
AHMAD MOLLAKHALILI
Affiliation:
Department of Mathematics, Kharazmi University (Tarbiat Moallem University), 50, Taleghani Ave., 15618, Tehran, Iran e-mail: [email protected], [email protected]
MATTHIAS NEUFANG
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, CanadaK1S 5B6, and Université Lille 1 - Sciences et Technologies, U.F.R. de Mathématiques, Laboratoire de Mathématiques Paul Painlevé - UMR CNRS 8524, 59655 Villeneuve d'Ascq Cédex, France e-mail: [email protected]
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.

A locally compact group G is compact if and only if its convolution algebras contain non-zero (weakly) completely continuous elements. Dually, G is discrete if its function algebras contain non-zero completely continuous elements. We prove non-commutative versions of these results in the case of locally compact quantum groups.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Baaj, S., Skandalis, G. and Vaes, S., Non-semi-regular quantum groups coming from number theory, Comm. Math. Phys. 235 (2003), 139167.Google Scholar
2. Bedos, E. and Tuset, L., Amenability and co-amenability for locally compact quantum groups, Internat. J. Math. 14 (2003), 865884.CrossRefGoogle Scholar
3. Berglund, J. F., Junghenn, H. D. and Milnes, P., Analysis on semigroups. Function spaces, compactifications, representations, Canadian Mathematical Society Series of Monographs and Advanced Texts (John Wiley & Sons, Inc., New York, 1989).Google Scholar
4. Brešar, M. and Turovskii, Y. V., Compactness conditions for elementary operators, Studia Math. 178 (1) (2007), 118.Google Scholar
5. Daws, M., Remarks on the quantum Bohr compactification, Illinois J. Math., 57 (4) (2013), 11311171.CrossRefGoogle Scholar
6. Daws, M., Fima, P., Skalski, A. and White, S., The Haagerup property for locally compact quantum groups, J. Reine Angew. Math., 711 (2016), 189229.Google Scholar
7. Diestel, J. and Uhl, J., Vector measures (American Mathematical Society, Providence, RI, 1977).CrossRefGoogle Scholar
8. Effros, E. and Ruan, Z.-J., Discrete quantum groups I, the Haar measure, Internat. J. Math. 5 (1994), 681723.Google Scholar
9. Erdos, J. A., On certain elements of C*-algebras, Illinois J. Math. 15 (1971), 682693.Google Scholar
10. Ghahramani, F. and Lau, A. T., Multipliers and ideals in second conjugate algebras related to locally compact groups, J. Funct. Anal. 132 (1995), 170191.CrossRefGoogle Scholar
11. Hu, Z., Neufang, M. and Ruan, Z.-J., Convolution of trace class operators over locally compact quantum groups, Canad. J. Math. 65 (5) (2013), 10431072.Google Scholar
12. Hu, Z., Neufang, M. and Ruan, Z.-J., Module maps over locally compact quantum groups, Studia Math. 211 (2) (2012), 111145.Google Scholar
13. Kalantar, M., Compact operators in regular LCQ groups, Canad. Math. Bull. 57 (3) (2014), 546550.Google Scholar
14. Kalantar, M., Towards harmonic analysis on locally compact quantum groups from groups to quantum groups – and back, PhD Thesis (Carleton University, 2011).Google Scholar
15. Kalantar, M. and Neufang, M., From quantum groups to groups, Canad. J. Math. 65 (5) (2013), 10731094.Google Scholar
16. Kustermans, J. and Vaes, S., Locally compact quantum groups, Ann. Sci. Èole Norm. Sup. 33 (2000), 837934.CrossRefGoogle Scholar
17. Kustermans, J. and Vaes, S., Locally compact quantum groups in the Von Neumann algebraic setting, Math. Scand. 92 (2003), 6892.CrossRefGoogle Scholar
18. Lau, A. T.-M. and Losert, V., On the second conjugate algebra of L 1(G) of a locally compact group, J. London Math. Soc. 37 (2) (1988), 464470.Google Scholar
19. Losert, V., Weakly compact multipliers on group algebras, J. Funct. Anal. 213 (2004), 466472.Google Scholar
20. Murphy, G.J., C*-algebras and operator theory (Academic Press, Inc., San Diego, CA, 1990).Google Scholar
21. Neufang, M., Solution to Farhadi–Ghahramani's multiplier problem, Proc. Amer. Math. Soc. 138 (2) (2010), 553555.Google Scholar
22. Palmer, T. W., Banach algebras and general theory of *-algebras, vol. 1 (Cambridge University Press, Cambridge, 1994).Google Scholar
23. Runde, V., Characterizations of compact and discrete quantum groups through second duals, J. Operator Theory 60 (2008), 415428.Google Scholar
24. Runde, V., Completely almost periodic functionals, Arch. Math. (Basel) 97 (2011), 325331.Google Scholar
25. Runde, V., Uniform continuity over locally compact quantum groups, J. London Math. Soc. 80 (2009), 5571.Google Scholar
26. Sakai, S., Weakly compact operators on operator algebras, Pacific J. Math. 14 (1964), 659664.Google Scholar
27. Sołtan, P.M., Quantum Bohr compactification, Illinois J. Math. 49 (2005), 12451270.Google Scholar
28. Spinu, E., Operator ideals on ordered Banach spaces, PhD Thesis (University of Alberta, 2013).Google Scholar
29. Takesaki, M., Theory of operator algebras, vol. 1 (Springer-Verlag, New York-Heidelberg, 1979).Google Scholar
30. Taylor, K. F., Geometry of the Fourier algebras and locally compact groups with atomic unitary representations, Math. Ann. 262 (2) (1983), 183190.Google Scholar
31. Woronowicz, S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (4) (1987), 613665.Google Scholar
32. Ylinen, K., A note on the compact elements of C*-algebras, Proc. Amer. Math. Soc. 35 (1972), 305306.Google Scholar
33. Ylinen, K., Weakly completely continuous elements of C*-algebras, Proc. Amer. Math. Soc. 52 (1975), 323326.Google Scholar