On hereditary properties of quantum group amenability

Crann, Jason

doi:10.1090/proc/13365

On hereditary properties of quantum group amenability
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by Jason Crann

Proc. Amer. Math. Soc. 145 (2017), 627-635

DOI: https://doi.org/10.1090/proc/13365

Published electronically: October 18, 2016

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Abstract:

Given a locally compact quantum group $\mathbb {G}$ and a closed quantum subgroup $\mathbb {H}$, we show that $\mathbb {G}$ is amenable if and only if $\mathbb {H}$ is amenable and $\mathbb {G}$ acts amenably on the quantum homogenous space $\mathbb {G}/\mathbb {H}$. We also study the existence of $L^1(\widehat {\mathbb {G}})$-module projections from $L^{\infty }(\widehat {\mathbb {G}})$ onto $L^{\infty }(\widehat {\mathbb {H}})$.

References

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
Man Duen Choi, A Schwarz inequality for positive linear maps on $C^{\ast } \$-algebras, Illinois J. Math. 18 (1974), 565–574. MR 355615
J. Crann, Amenability and covariant injectivity of locally compact quantum groups II. Canadian J. Math. to appear. arXiv:1507.03296.
Jason Crann and Matthias Neufang, Amenability and covariant injectivity of locally compact quantum groups, Trans. Amer. Math. Soc. 368 (2016), no. 1, 495–513. MR 3413871, DOI 10.1090/tran/6374
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
Matthew Daws, Categorical aspects of quantum groups: multipliers and intrinsic groups, Canad. J. Math. 68 (2016), no. 2, 309–333. MR 3484369, DOI 10.4153/CJM-2015-022-0
Matthew Daws, PawełKasprzak, Adam Skalski, and Piotr M. Sołtan, Closed quantum subgroups of locally compact quantum groups, Adv. Math. 231 (2012), no. 6, 3473–3501. MR 2980506, DOI 10.1016/j.aim.2012.09.002
Jacques Delaporte and Antoine Derighetti, Invariant projections and convolution operators, Proc. Amer. Math. Soc. 129 (2001), no. 5, 1427–1435. MR 1814169, DOI 10.1090/S0002-9939-00-05874-3
Antoine Derighetti, Conditional expectations on $CV_p(G)$. Applications, J. Funct. Anal. 247 (2007), no. 1, 231–251. MR 2319760, DOI 10.1016/j.jfa.2007.03.003
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 Akitaka Kishimoto, Module maps and Hochschild-Johnson cohomology, Indiana Univ. Math. J. 36 (1987), no. 2, 257–276. MR 891774, DOI 10.1512/iumj.1987.36.36015
B. Forrest, E. Kaniuth, A. T. Lau, and N. Spronk, Ideals with bounded approximate identities in Fourier algebras, J. Funct. Anal. 203 (2003), no. 1, 286–304. MR 1996874, DOI 10.1016/S0022-1236(02)00121-0
Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482
Mehrdad Kalantar and Matthias Neufang, From quantum groups to groups, Canad. J. Math. 65 (2013), no. 5, 1073–1094. MR 3095007, DOI 10.4153/CJM-2012-047-x
Eberhard Kaniuth and Anthony T. Lau, A separation property of positive definite functions on locally compact groups and applications to Fourier algebras, J. Funct. Anal. 175 (2000), no. 1, 89–110. MR 1774852, DOI 10.1006/jfan.2000.3612
PawełKasprzak and Piotr M. Sołtan, Embeddable quantum homogeneous spaces, J. Math. Anal. Appl. 411 (2014), no. 2, 574–591. MR 3128415, DOI 10.1016/j.jmaa.2013.07.084
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
Horst Leptin, Sur l’algèbre de Fourier d’un groupe localement compact, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A1180–A1182 (French). MR 239002
S. Vaes, Locally Compact Quantum Groups. Ph.D. thesis, K.U. Leuven, 2000.
Stefaan Vaes, A new approach to induction and imprimitivity results, J. Funct. Anal. 229 (2005), no. 2, 317–374. MR 2182592, DOI 10.1016/j.jfa.2004.11.016
Alfons Van Daele, Locally compact quantum groups. A von Neumann algebra approach, SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 082, 41. MR 3261862, DOI 10.3842/SIGMA.2014.082