Amenability and covariant injectivity of locally compact quantum groups

Crann, Jason; Neufang, Matthias

doi:10.1090/tran/6374

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