Amenability and covariant injectivity of locally compact quantum groups
HTML articles powered by AMS MathViewer
- by Jason Crann and Matthias Neufang PDF
- Trans. Amer. Math. Soc. 368 (2016), 495-513 Request permission
Abstract:
As is well known, the equivalence between amenability of a locally compact group $G$ and injectivity of its von Neumann algebra $\mathcal {L}(G)$ does not hold in general beyond inner amenable groups. In this paper, we show that the equivalence persists for all locally compact groups if $\mathcal {L}(G)$ is considered as a $\mathcal {T}(L_2(G))$-module with respect to a natural action. In fact, we prove an appropriate version of this result for every locally compact quantum group.References
- Oleg Yu. Aristov, Amenability and compact type for Hopf-von Neumann algebras from the homological point of view, Banach algebras and their applications, Contemp. Math., vol. 363, Amer. Math. Soc., Providence, RI, 2004, pp. 15–37. MR 2097947, DOI 10.1090/conm/363/06638
- E. Bédos and L. Tuset, Amenability and co-amenability for locally compact quantum groups, Internat. J. Math. 14 (2003), no. 8, 865–884. MR 2013149, DOI 10.1142/S0129167X03002046
- J. Crann and M. Neufang, Inner amenable locally compact quantum groups. preprint (2013).
- Erik Christensen and Allan M. Sinclair, On von Neumann algebras which are complemented subspaces of $B(H)$, J. Funct. Anal. 122 (1994), no. 1, 91–102. MR 1274585, DOI 10.1006/jfan.1994.1063
- A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
- H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1816726
- H. G. Dales and M. E. Polyakov, Homological properties of modules over group algebras, Proc. London Math. Soc. (3) 89 (2004), no. 2, 390–426. MR 2078704, DOI 10.1112/S0024611504014686
- Pieter Desmedt, Johan Quaegebeur, and Stefaan Vaes, Amenability and the bicrossed product construction, Illinois J. Math. 46 (2002), no. 4, 1259–1277. MR 1988262
- Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
- Edmond E. Granirer, Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 189 (1974), 371–382. MR 336241, DOI 10.1090/S0002-9947-1974-0336241-0
- U. Haagerup, Decomposition of completely bounded maps on operator algebras. Unpublished results, Odense University, Denmark, 1980.
- A. Ya. Helemskii, Banach and locally convex algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. Translated from the Russian by A. West. MR 1231796
- Zhiguo Hu, Matthias Neufang, and Zhong-Jin Ruan, Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres, Proc. Lond. Math. Soc. (3) 100 (2010), no. 2, 429–458. MR 2595745, DOI 10.1112/plms/pdp026
- Zhiguo Hu, Matthias Neufang, and Zhong-Jin Ruan, Completely bounded multipliers over locally compact quantum groups, Proc. Lond. Math. Soc. (3) 103 (2011), no. 1, 1–39. MR 2812500, DOI 10.1112/plms/pdq041
- Zhiguo Hu, Matthias Neufang, and Zhong-Jin Ruan, Module maps over locally compact quantum groups, Studia Math. 211 (2012), no. 2, 111–145. MR 2997583, DOI 10.4064/sm211-2-2
- Marius Junge, Matthias Neufang, and Zhong-Jin Ruan, A representation theorem for locally compact quantum groups, Internat. J. Math. 20 (2009), no. 3, 377–400. MR 2500076, DOI 10.1142/S0129167X09005285
- Mehrdad Kalantar and Matthias Neufang, Duality, cohomology, and geometry of locally compact quantum groups, J. Math. Anal. Appl. 406 (2013), no. 1, 22–33. MR 3062398, DOI 10.1016/j.jmaa.2013.04.024
- Johan Kustermans and Stefaan Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 837–934 (English, with English and French summaries). MR 1832993, DOI 10.1016/S0012-9593(00)01055-7
- Johan Kustermans and Stefaan Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), no. 1, 68–92. MR 1951446, DOI 10.7146/math.scand.a-14394
- Anthony To Ming Lau and Alan L. T. Paterson, Inner amenable locally compact groups, Trans. Amer. Math. Soc. 325 (1991), no. 1, 155–169. MR 1010885, DOI 10.1090/S0002-9947-1991-1010885-5
- G. May, E. Neuhardt, and G. Wittstock, The space of completely bounded module homomorphisms, Arch. Math. (Basel) 53 (1989), no. 3, 283–287. MR 1006722, DOI 10.1007/BF01277066
- M. Neufang, Abstrakte Harmonische Analyse und Modulhomomorphismen über von Neumann-Algebren. PhD thesis, University of Saarland, 2000.
- Matthias Neufang, Zhong-Jin Ruan, and Nico Spronk, Completely isometric representations of $M_{cb}A(G)$ and $UCB(\hat G)^\ast$, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1133–1161. MR 2357691, DOI 10.1090/S0002-9947-07-03940-2
- Matthias Neufang and Volker Runde, Harmonic operators: the dual perspective, Math. Z. 255 (2007), no. 3, 669–690. MR 2270293, DOI 10.1007/s00209-006-0039-6
- Gilles Pisier, The operator Hilbert space $\textrm {OH}$, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103. MR 1342022, DOI 10.1090/memo/0585
- A. Yu. Pirkovskii, Biprojectivity and biflatness for convolution algebras of nuclear operators, Canad. Math. Bull. 47 (2004), no. 3, 445–455. MR 2072605, DOI 10.4153/CMB-2004-044-6
- P. F. Renaud, Invariant means on a class of von Neumann algebras, Trans. Amer. Math. Soc. 170 (1972), 285–291. MR 304553, DOI 10.1090/S0002-9947-1972-0304553-0
- Zhong-Jin Ruan, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139 (1996), no. 2, 466–499. MR 1402773, DOI 10.1006/jfan.1996.0093
- Volker Runde, Uniform continuity over locally compact quantum groups, J. Lond. Math. Soc. (2) 80 (2009), no. 1, 55–71. MR 2520377, DOI 10.1112/jlms/jdp011
- Piotr M. Sołtan and Ami Viselter, A note on amenability of locally compact quantum groups, Canad. Math. Bull. 57 (2014), no. 2, 424–430. MR 3194189, DOI 10.4153/CMB-2012-032-3
- Masamichi Takesaki, A characterization of group algebras as a converse of Tannaka-Stinespring-Tatsuuma duality theorem, Amer. J. Math. 91 (1969), 529–564. MR 244437, DOI 10.2307/2373525
- M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6. MR 1943006, DOI 10.1007/978-3-662-10451-4
- J. Tomiyama, Tensor products and properties of projections of norm one in von Neumann algebras. Unpublished Lecture Notes, University of Copenghagen, 1970.
- Peter James Wood, Homological algebra in operator spaces with applications to harmonic analysis, ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)–University of Waterloo (Canada). MR 2699432
- S. Vaes, Locally compact quantum groups. PhD thesis, Katholieke Universiteit, 2000.
- Stefaan Vaes and Leonid Vainerman, On low-dimensional locally compact quantum groups, Locally compact quantum groups and groupoids (Strasbourg, 2002) IRMA Lect. Math. Theor. Phys., vol. 2, de Gruyter, Berlin, 2003, pp. 127–187. MR 1976945
- A. Van Daele, Locally compact quantum groups. A von Neumann algebra approach. preprint arXiv:math/0602212 (2006).
- Amin Zobeidi, Every topologically amenable locally compact quantum group is amenable, Bull. Aust. Math. Soc. 87 (2013), no. 1, 149–151. MR 3011950, DOI 10.1017/S0004972712000275
Additional Information
- Jason Crann
- Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6 – and – Université Lille 1 - Sciences et Technologies, UFR de Mathématiques, Laboratoire de Mathématiques Paul Painlevé - UMR CNRS 8524, 59655 Villeneuve d’Ascq Cédex, France
- Email: jason_crann@carleton.ca
- Matthias Neufang
- Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6 – and – Université Lille 1 - Sciences et Technologies, UFR de Mathématiques, Laboratoire de Mathématiques Paul Painlevé - UMR CNRS 8524, 59655 Villeneuve d’Ascq Cédex, France
- MR Author ID: 718390
- Email: Matthias.Neufang@carleton.ca
- Received by editor(s): April 15, 2013
- Received by editor(s) in revised form: November 19, 2013
- Published electronically: May 22, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 495-513
- MSC (2010): Primary 22D15, 46L89, 81R15; Secondary 43A07, 46M10, 43A20
- DOI: https://doi.org/10.1090/tran/6374
- MathSciNet review: 3413871