Inner amenability and approximation properties of locally compact quantum groups. (English) Zbl 1464.46072
Summary: We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau-Paterson [A. T. M. Lau and A. L. T. Paterson, Trans. Am. Math. Soc. 325, No. 1, 155–169 (1991; Zbl 0718.43002)],
resolve a recent conjecture of Ng-Viselter [C.-K. Ng and A. Viselter, Bull. Lond. Math. Soc. 49, No. 3, 491–498 (2017; Zbl 1439.46059)],
and prove that, for inner amenable quantum groups \(\mathbb{G}\), approximation properties of the dual operator algebras can be averaged to approximation properties of \(\mathbb{G}\). Similar homological techniques are used to prove that \(\ell^1(\mathbb{G})\) is not relatively operator biflat for any non-unimodular discrete quantum group \(\mathbb{G}\); a unimodular discrete quantum group \(\mathbb{G}\) with Kirchberg’s factorization property is weakly amenable if and only if \(L^1_\text{cb}(\widehat{\mathbb{G}})\) is operator amenable, and amenability of a locally compact quantum group \(\mathbb{G}\) implies \(C_u(\widehat{\mathbb{G}}) \cong L^1 (\widehat{\mathbb{G}}) \widehat{\otimes}_{L^1(\widehat{\mathbb{G}})} C_0(\widehat{\mathbb{G}})\) completely isometrically. The latter result allows us to partially answer a conjecture of Voiculescu [D. Voiculescu, in: Algèbres d’opérateurs et leurs applications en physique mathématique, Colloq. int. CNRS No. 274, Marseille 1977, 451–457 (1979; Zbl 0503.46049)] when \(\mathbb{G}\) has the approximation property.
MSC:
| 46L67 | Quantum groups (operator algebraic aspects) |
| 46M10 | Projective and injective objects in functional analysis |
| 22D35 | Duality theorems for locally compact groups |
| 46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |
References:
| [1] | C.ANANTHARAMAN-DELAROCHE,Amenability and exactness for dynamical systems and their C∗-algebras, Trans. Amer. Math. Soc.354(2002), no. 10, 4153-4178.http://dx.doi.org/ 10.1090/S0002-9947-02-02978-1.MR1926869 · Zbl 1035.46039 |
| [2] | O. 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.http://dx.doi.org/10.1090/conm/363/ 06638.MR2097947 · Zbl 1076.46042 |
| [3] | ,Fourier algebras of some connected groups are not projective, Uspekhi Mat. Nauk60 (2005), no. 1(361), 159-160 (Russian); English transl., Russian Math. Surveys60(2005), no. 1, 154-156.http://dx.doi.org/10.1070/RM2005v060n01ABEH000811.MR2145663 · Zbl 1079.43005 |
| [4] | O. Yu.ARISTOV, V.RUNDE, and N.SPRONK,Operator biflatness of the Fourier algebra and approximate indicators for subgroups, J. Funct. Anal.209(2004), no. 2, 367-387.http://dx. doi.org/10.1016/S0022-1236(03)00169-1.MR2044227 · Zbl 1052.22005 |
| [5] | E.B ´EDOSand L.TUSET,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.MR2013149 · Zbl 1051.46047 |
| [6] | M. B.BEKKA,On the fullC∗-algebras of arithmetic groups and the congruence subgroup problem, Forum Math.11(1999), no. 6, 705-715.http://dx.doi.org/10.1515/form.1999. 021.MR1725593 · Zbl 0940.22003 |
| [7] | A.BHATTACHARYAand S.WANG,Kirchberg’s factorization property for discrete quantum groups, Bull. Lond. Math. Soc.48(2016), no. 5, 866-876.http://dx.doi.org/10.1112/blms/ bdw048.MR3556369 · Zbl 1364.46062 |
| [8] | D. P.BLECHER,The standard dual of an operator space, Pacific J. Math.153(1992), no. 1, 15- 30.MR1145913 · Zbl 0726.47030 |
| [9] | D. P.BLECHERand Ch.LEMERDY,Operator Algebras and their Modules—an Operator Space Approach, London Mathematical Society Monographs. New Series. Oxford Science Publications, vol. 30, The Clarendon Press, Oxford University Press, Oxford, 2004.http://dx.doi.org/10. 1093/acprof:oso/9780198526599.001.0001.MR2111973 · Zbl 1061.47002 |
| [10] | M.BRANNAN, B.COLLINS, and R.VERGNIOUX,The Connes embedding property for quantum group von Neumann algebras, Trans. Amer. Math. Soc.369(2017), no. 6, 3799-3819.http:// dx.doi.org/10.1090/tran/6752.MR3624393 · Zbl 1371.46057 |
| [11] | N. P.BROWNand K. J.DYKEMA,Popa algebras in free group factors, J. Reine Angew. Math.573 (2004), 157-180.http://dx.doi.org/10.1515/crll.2004.058.MR2084586 · Zbl 1068.46030 |
| [12] | J.DECANNIERE‘and U.HAAGERUP,Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math.107(1985), no. 2, 455-500.http://dx.doi.org/ 10.2307/2374423.MR784292 · Zbl 0577.43002 |
| [13] | M.CASPERS, H.H.LEE, and ´E.RICARD,Operator biflatness of theL1-algebras of compact quantum groups, J. Reine Angew. Math.700(2015), 235-244.http://dx.doi.org/10.1515/ crelle-2013-0016.MR3318517 · Zbl 1322.46036 |
| [14] | J.CRANN,Homological manifestations of quantum group duality, PhD Thesis, Carleton University, 2015. |
| [15] | ,Amenability and covariant injectivity of locally compact quantum groups II, Canad.J.Math.69(2017),no.5,1064-1086.http://dx.doi.org/10.4153/ CJM-2016-031-5.MR3693148 · Zbl 1377.22006 |
| [16] | J.CRANNand M.NEUFANG,Amenability and covariant injectivity of locally compact quantum groups, Trans. Amer. Math. Soc.368(2016), no. 1, 495-513. http://dx.doi.org/10.1090/tran/6374.MR3413871 · Zbl 1330.22013 |
| [17] | J.CRANNand Z.TANKO,On the operator homology of the Fourier algebra and itscb-multiplier completion, J. Funct. Anal.273(2017), no. 7, 2521-2545.http://dx.doi.org/10.1016/j. jfa.2017.06.024.MR3677832 · Zbl 1384.46035 |
| [18] | A.VANDAELE,Locally compact quantum groups:A von Neumann algebra approach, SIGMA Symmetry Integrability Geom. Methods Appl.10(2014), Paper 082, 41.http://dx.doi.org/ 10.3842/SIGMA.2014.082.MR3261862 · Zbl 1312.46055 |
| [19] | H. G.DALESand M. E.POLYAKOV,Homological properties of modules over group algebras, Proc. London Math. Soc. (3)89(2004), no. 2, 390-426.http://dx.doi.org/10.1112/ S0024611504014686.MR2078704 · Zbl 1058.43003 |
| [20] | M.DAWS,Multipliers, self-induced and dual Banach algebras, Dissertationes Math. (Rozprawy Mat.)470(2010), 62.http://dx.doi.org/10.4064/dm470-0-1.MR2681109 · Zbl 1214.43004 |
| [21] | ,Multipliers of locally compact quantum groups via HilbertC∗-modules, J. Lond. Math. Soc. (2)84(2011), no. 2, 385-407. http://dx.doi.org/10.1112/jlms/jdr013.MR2835336 · Zbl 1235.43004 |
| [22] | ,Completely positive multipliers of quantum groups, Internat. J. Math.23(2012), no. 12, 1250132, 23.http://dx.doi.org/10.1142/S0129167X12501327.MR3019431 · Zbl 1282.43002 |
| [23] | ,Categorical aspects of quantum groups: multipliers and intrinsic groups, Canad. J. Math. 68(2016), no. 2, 309-333.http://dx.doi.org/10.4153/CJM-2015-022-0.MR3484369 · Zbl 1355.46061 |
| [24] | M.DAWS, P.FIMA, A.SKALSKI, and S.WHITE,The Haagerup property for locally compact quantum groups, J. Reine Angew. Math.711(2016), 189-229.http://dx.doi.org/10.1515/ crelle-2013-0113.MR3456763 · Zbl 1343.46068 |
| [25] | M.DAWS, P.KASPRZAK, A.SKALSKI, and P. M.SOŁTAN,Closed quantum subgroups of locally compact quantum groups, Adv. Math.231(2012), no. 6, 3473-3501.http://dx.doi.org/10. 1016/j.aim.2012.09.002.MR2980506 · Zbl 1275.46057 |
| [26] | M.DAWSand P.SALMI,Completely positive definite functions and Bochner’s theorem for locally compact quantum groups, J. Funct. Anal.264(2013), no. 7, 1525-1546.http://dx.doi.org/ 10.1016/j.jfa.2013.01.017.MR3019722 · Zbl 1320.46056 |
| [27] | P.DESMEDT, J.QUAEGEBEUR, and S.VAES,Amenability and the bicrossed product construction, Illinois J. Math.46(2002), no. 4, 1259-1277.MR1988262 · Zbl 1035.46042 |
| [28] | E. G.EFFROS,PropertyΓand inner amenability, Proc. Amer. Math. Soc.47(1975), 483-486. http://dx.doi.org/10.2307/2039768.MR0355626 · Zbl 0321.22011 |
| [29] | E. G.EFFROSand Z. J.RUAN,Operator Spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000.MR1793753 · Zbl 0969.46002 |
| [30] | B. E.FORREST, V.RUNDE, and N.SPRONK,Operator amenability of the Fourier algebra in the cb-multiplier norm, Canad. J. Math.59(2007), no. 5, 966-980.http://dx.doi.org/10.4153/ CJM-2007-041-9.MR2354398 · Zbl 1136.43002 |
| [31] | B. E.FORREST,N. SPRONK, and M.WIERSMA,Existence of tracial states on reduced group C∗-algebras(2017), preprint, available athttp://arxiv.org/abs/arXiv:1706.05354. |
| [32] | A.FRESLON,Examples of weakly amenable discrete quantum groups, J. Funct. Anal.265(2013), no. 9, 2164-2187.http://dx.doi.org/10.1016/j.jfa.2013.05.037.MR3084500 · Zbl 1328.46064 |
| [33] | ,A note on weak amenability for free products of discrete quantum groups, C. R. Math. Acad. Sci. Paris350(2012), no. 7-8, 403-406 (English, with English and French summaries). http://dx.doi.org/10.1016/j.crma.2012.04.015.MR2922092 · Zbl 1252.46058 |
| [34] | M. R.GHANEIand R.NASR-ISFAHANI,Inner amenability of locally compact quantum groups, Internat. J. Math.24(2013), no. 7, 1350058, 17.http://dx.doi.org/10.1142/ S0129167X13500584.MR3084739 · Zbl 1288.43001 |
| [35] | U. Haagerup and J. Kraus,Approximation properties for groupC∗-algebras and group von Neumann algebras, Trans. Amer. Math. Soc.44(1994), no. 2, 667-699.http://dx.doi.org/10.2307/ 2154501.MR1220905 · Zbl 0806.43002 |
| [36] | Z.HU, M.NEUFANG, and Z. J.RUAN,Completely bounded multipliers over locally compact quantum groups, Proc. Lond. Math. Soc. (3)103(2011), no. 1, 1-39.http://dx.doi.org/10. 1112/plms/pdq041.MR2812500 · Zbl 1250.22005 |
| [37] | B. E.JOHNSON,Cohomology in Banach Algebras, Memoirs of the American Mathematical Society, vol. 127, American Mathematical Society, Providence, R.I., 1972.MR0374934 · Zbl 0256.18014 |
| [38] | M.JUNGE, M.NEUFANG, and Z. J.RUAN,A representation theorem for locally compact quantum groups, Internat. J. Math.20(2009), no. 3, 377-400.http://dx.doi.org/10.1142/ S0129167X09005285.MR2500076 · Zbl 1194.22003 |
| [39] | M.KALANTARand M.NEUFANG,Duality, cohomology, and geometry of locally compact quantum groups, J. Math. Anal. Appl.406(2013), no. 1, 22-33.http://dx.doi.org/10.1016/j.jmaa. 2013.04.024.MR3062398 · Zbl 1310.58003 |
| [40] | J. L.KELLEY,General Topology, Reprint of the 1955 edition [Van Nostrand, Toronto, Ont., Graduate Texts in Mathematics, vol. 27, Springer-Verlag, New York-Berlin, 1975.MR0370454 · Zbl 0306.54002 |
| [41] | M.KENNEDYand S.RAUM,Traces on reduced groupC∗-algebras, Bull. Lond. Math. Soc.49 (2017), no. 6, 988-990, available athttp://arxiv.org/abs/arXiv:1706.05903.http://dx. doi.org/10.1112/blms.12101.MR3743482 · Zbl 1382.22010 |
| [42] | J.KRAUSand Z. J.RUAN,Approximation properties for Kac algebras, Indiana Univ. Math. J.48 (1999), no. 2, 469-535.http://dx.doi.org/10.1512/iumj.1999.48.1660.MR1722805 · Zbl 0945.46038 |
| [43] | J.KUSTERMANS,Locally compact quantum groups in the universal setting, Internat. J. Math.12 (2001), no. 3, 289-338.http://dx.doi.org/10.1142/S0129167X01000757.MR1841517 · Zbl 1111.46311 |
| [44] | J.KUSTERMANSand S.VAES,Locally compact quantum groups, Ann. Sci. ´Ecole Norm. Sup. (4)33(2000), no. 6, 837-934 (English, with English and French summaries).http://dx.doi. org/10.1016/S0012-9593(00)01055-7.MR1832993 · Zbl 1034.46508 |
| [45] | ,Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand.92 (2003), no. 1, 68-92.http://dx.doi.org/10.7146/math.scand.a-14394.MR1951446 · Zbl 1034.46067 |
| [46] | E. C.LANCE,HilbertC∗-modules:A Toolkit for Operator Algebraists, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995.http://dx. doi.org/10.1017/CBO9780511526206.MR1325694 · Zbl 0822.46080 |
| [47] | A. T. M.LAUand A. L. T.PATERSON,Inner amenable locally compact groups, Trans. Amer. Math. Soc.325(1991), no. 1, 155-169.http://dx.doi.org/10.2307/2001664.MR1010885 · Zbl 0718.43002 |
| [48] | H.LEPTIN,Sur l’alg‘ebre de Fourier d’un groupe localement compact, C. R. Acad. Sci. Paris S´er. A-B266(1968), A1180-A1182 (French).MR0239002 · Zbl 0169.46501 |
| [49] | V.LOSERTand H.RINDLER,Asymptotically central functions and invariant extensions of Dirac measure, Probability Measures on Groups, VII, Oberwolfach (1983), Lecture Notes in Math., vol. 1064, Springer, Berlin, 1984, pp. 368-378.http://dx.doi.org/10.1007/ BFb0073651.MR772419 · Zbl 0573.43003 |
| [50] | S.MAJID,Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts, and the classical Yang-Baxter equations, J. Funct. Anal.95(1991), no. 2, 291-319.http://dx.doi.org/10. 1016/0022-1236(91)90031-Y.MR1092128 · Zbl 0741.46033 |
| [51] | C. K.NG,Strictly amenable representations of reduced groupC∗-algebras, Int. Math. Res. Not. IMRN17(2015), 7853-7860.http://dx.doi.org/10.1093/imrn/rnu183.MR3404003 · Zbl 1332.22009 |
| [52] | C. K.NGand A.VISELTER,Amenability of locally compact quantum groups and their unitary co-representations, Bull. Lond. Math. Soc.49(2017), no. 3, 491-498.http://dx.doi.org/10. 1112/blms.12047.MR3723633 · Zbl 1439.46059 |
| [53] | R.OKAYASU, N.OZAWA, and R.TOMATSU,Haagerup approximation property via bimodules, Math. Scand.121(2017), no. 1, 75-91, available athttp://arxiv.org/abs/arXiv:1501. 06293.http://dx.doi.org/10.7146/math.scand.a-25970.MR3708965 · Zbl 1430.46043 |
| [54] | A. L. T.PATERSON,Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988.http://dx.doi.org/10.1090/surv/ 029.MR961261 · Zbl 0648.43001 |
| [55] | Z. J.RUAN,The operator amenability ofA(G), Amer. J. Math.117(1995), no. 6, 1449-1474. http://dx.doi.org/10.2307/2375026.MR1363075 · Zbl 0842.43004 |
| [56] | ,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.MR1402773 · Zbl 0896.46041 |
| [57] | Z. J.RUANand G.XU,Splitting properties of operator bimodules and operator amenability of Kac algebras, Operator Theory, Operator Algebras and Related Topics, Timis¸oara (1996), Theta Found., Bucharest, 1997, pp. 193-216.MR1728421 · Zbl 0942.46033 |
| [58] | V.RUNDEand A.VISELTER,On positive definiteness over locally compact quantum groups, Canad. J. Math.68(2016), no. 5, 1067-1095.http://dx.doi.org/10.4153/ CJM-2015-019-0.MR3536928 · Zbl 1352.22004 |
| [59] | R.STOKKE,Quasi-central bounded approximate identities in group algebras of locally compact groups, Illinois J. Math.48(2004), no. 1, 151-170.MR2048220 · Zbl 1037.43004 |
| [60] | Y.SUZUKI,Elementary constructions of non-discreteC∗-simple groups, Proc. Amer. Math. Soc. 145(2017), no. 3, 1369-1371.http://dx.doi.org/10.1090/proc/13301.MR3589332 · Zbl 1354.22010 |
| [61] | M.TAKESAKI,A characterization of group algebras as a converse of Tannaka-StinespringTatsuuma duality theorem, Amer. J. Math.91(1969), 529-564.http://dx.doi.org/10.2307/ 2373525.MR0244437 · Zbl 0182.18103 |
| [62] | ,Theory of Operator Algebras. I, Reprint of the first (1979) edition, Encyclopaedia of Mathematical Sciences: Operator Algebras and Non-commutative Geometry, 5, vol. 124, Springer-Verlag, Berlin, 2002.MR1873025 · Zbl 0990.46034 |
| [63] | ,Theory of Operator Algebras. II, Encyclopaedia of Mathematical Sciences: Operator Algebras and Non-commutative Geometry, 6, vol. 125, Springer-Verlag, Berlin, 2003.http://dx. doi.org/10.1007/978-3-662-10451-4.MR1943006 · Zbl 1059.46031 |
| [64] | S.VAES,The unitary implementation of a locally compact quantum group action, J. Funct. Anal. 180(2001), no. 2, 426-480.http://dx.doi.org/10.1006/jfan.2000.3704.MR1814995 · Zbl 1011.46058 |
| [65] | ,A new approach to induction and imprimitivity results, J. Funct. Anal.229(2005), no. 2, 317-374.http://dx.doi.org/10.1016/j.jfa.2004.11.016.MR2182592 · Zbl 1087.22005 |
| [66] | S.VAESand L.VAINERMAN,Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math.175(2003), no. 1, 1-101.http://dx.doi.org/10.1016/ S0001-8708(02)00040-3.MR1970242 · Zbl 1034.46068 |
| [67] | D.VOICULESCU,Amenability and Katz algebras, Alg‘ebres d’op´erateurs et leurs applications en physique math´ematique, Proc. Colloq., Marseille, (1977), Colloq. Internat. CNRS, vol. 274, CNRS, Paris, 1979, pp. 451-457.MR560656 |
| [68] | S. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
