×

zbMATH — the first resource for mathematics

Compact and weakly compact multipliers of locally compact quantum groups. (English) Zbl 1417.46047
Summary: A locally compact group \(G\) is compact if and only if its convolution algebra has a non-zero (weakly) compact multiplier. Dually, \(G\) is discrete if and only if its Fourier algebra has a non-zero (weakly) compact multiplier. In addition, \(G\) is compact (respectively, amenable) if and only if the second dual of its convolution algebra equipped with the first Arens product has a non-zero (weakly) compact left (respectively, right) multiplier. We prove the non-commutative versions of these results in the case of locally compact quantum groups.
MSC:
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L51 Noncommutative measure and integration
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aiena, P.: Fredholm and Local Spectral Theory, with Application to Multipliers. Kluwer Academic Publishers, Dordrecht (2004) · Zbl 1077.47001
[2] Amini, M.; Kalantar, M.; Medghalchi, A.; Mollakhalili, A.; Neufang, M., Compact elements and operators of quantum groups, Glasgow Math. J., 59, 445-462, (2017) · Zbl 1370.46048
[3] Baaj, S.; Skandalis, G.; Vaes, S., Non-semi-regular quantum groups coming from number theory, Commun. Math. Phys., 235, 139-167, (2003) · Zbl 1029.46113
[4] Baker, J.; Lau, AT-M; Pym, JS, Module homomorphisms and topological centers associated with weakly sequentially complete Banach algebras, J. Funct. Anal., 158, 186-208, (1998) · Zbl 0911.46030
[5] Bartle, R.; Dunford, N.; Schwartz, J., Weak compactness and vector measures, Can. J. Math., 7, 289-305, (1955) · Zbl 0068.09301
[6] Bedos, E.; Tuset, L., Amenability and co-amenability for locally compact quantum groups, Int. J. Math., 14, 865-884, (2003) · Zbl 1051.46047
[7] Berglund, J.F., Junghenn, H.D., Milnes, P.: Analysis on semigroups. Function spaces, compactifications, representations. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1989) · Zbl 0727.22001
[8] Bonsall, F.F., Duncan, J.: Complete Normed Algebras. Springer, New York (1973) · Zbl 0271.46039
[9] Brešar, M.; Eremita, D., The lower socle and finite rank elementary operators, Commun. Algebra, 31, 1485-1497, (2003) · Zbl 1021.16023
[10] Brešar, M.; Turovskii, YV, Compactness conditions for elementary operators, Stud. Math., 178, 1-18, (2007) · Zbl 1117.47023
[11] Brešar, M., On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J., 33, 89-93, (1991) · Zbl 0731.47037
[12] Crann, J.; Neufang, M., Amenability and covariant injectivity of locally compact quantum groups, Trans. Am. Math. Soc., 368, 495-513, (2016) · Zbl 1330.22013
[13] Dalla, L.; Giotopoulos, S.; Katseli, N., The socle and finite-dimensionality of a semiprime Banach algebra, Stud. Math., 92, 201-204, (1989) · Zbl 0691.46036
[14] Dales, H.G.: Banach Algebras and Automatic Continuity, London Mathematical Society Monographs. New Series, vol. 24. Oxford University Press, New York (2000)
[15] Dales, HG; Lau, AT-M, The second duals of Beurling algebras, Mem. Am. Math. Soc., 177, vi+191, (2005) · Zbl 1075.43003
[16] Daws, M., Completely positive multipliers of quantum groups, Int. J. Math., 23, 1250132, (2012) · Zbl 1282.43002
[17] Daws, M., Multipliers of locally compact quantum groups via Hilbert C\(^*\)-modules, J. Lond. Math. Soc. (2), 84, 385-407, (2011) · Zbl 1235.43004
[18] Daws, M., Multipliers, self-induced and dual Banach algebras, Dissertationes Math., 470, 62, (2010) · Zbl 1214.43004
[19] Daws, M., Remarks on the quantum Bohr compactification, Ill. J. Math., 57, 1131-1171, (2013) · Zbl 1305.43006
[20] Daws, M.; Pham, LH, Isometries between quantum convolotion algebras, Q. J. Math., 64, 373-396, (2013) · Zbl 1275.46043
[21] Daws, M.; Salmi, P., Completely positive definite functions and Bochner’s theorem for locally compact quantum groups, J. Funct. Anal., 264, 1525-1546, (2013) · Zbl 1320.46056
[22] Desmedt, P.; Quaegebeur, J.; Vaes, S., Amenability and the bicrossed product construction, Ill. J. Math., 46, 1259-1277, (2002) · Zbl 1035.46042
[23] Diestel, J.; Faires, B., On vector measures, Trans. Am. Math. Soc., 198, 253-271, (1974) · Zbl 0297.46034
[24] Dunford, N., Schwartz, J.T.: Linear Operators, I: General Theory, Pure and Applied Mathematics, vol. 7. Interscience, New York (1958) · Zbl 0084.10402
[25] Enock, M., Schwartz, J.M.: Kac Algebras and Duality of Locally Compact Groups. Springer, Berlin (1992) · Zbl 0805.22003
[26] Erdos, JA, On certain elements of \(C^*\)-algebras, Ill. J. Math., 15, 682-693, (1971) · Zbl 0222.46042
[27] Eshaghi Gordji, M.; Hosseiniun, SAR, The fourth dual of Banach algebras, Ital. J. Pure Appl. Math., 24, 53-60, (2007)
[28] Eymard, P., L’ alg\({\mathbb{G}}rave{e}\)bre de Fourier d’un groupe localement compact, Bull. Soc. Math. France, 92, 181-236, (1964) · Zbl 0169.46403
[29] Filali, M., Finite-dimensional left ideals in some algebras associated with a locally compact group, Proc. Am. Math. Soc., 127, 2325-2333, (1999) · Zbl 0918.43003
[30] Fima, P., On locally compact quantum groups whose algebras are factors, J. Funct. Anal., 244, 78-94, (2007) · Zbl 1121.46056
[31] Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (2016) · Zbl 1342.43001
[32] Forrest, B., Arens regularity and discrete groups, Pac. J. Math., 151, 217-227, (1991) · Zbl 0746.43002
[33] Ghaffari, A., Module homomorphisms associated with Banach algebras, Taiwan. J. Math., 15, 1075-1088, (2011) · Zbl 1231.43002
[34] Ghaffari, A.; Medghalchi, A., The Socle and finite dimensionality of some Banach algebras, Proc. Indian Acad. Sci. Math. Sci., 115, 327-330, (2005) · Zbl 1098.46034
[35] Ghahramani, F.; Lau, AT, Isomorphisms and multipliers on second dual algebras of Banach algebras, Math. Proc. Camb. Philos. Soc., 111, 161-168, (1992) · Zbl 0818.46050
[36] Ghahramani, F.; Lau, AT, Multipliers and ideals in second conjugate algebras related to locally compact groups, J. Funct. Anal., 132, 170-191, (1995) · Zbl 0832.22007
[37] Ghahramani, F.; Loy, RJ; Willis, GA, Amenability and weak amenability of second conjugate Banach algebras, Proc. Am. Math. Soc., 124, 1489-1497, (1996) · Zbl 0851.46035
[38] Hu, Z.; Monfared, MS; Traynor, T., On character amenable Banach algebra, Stud. Math., 193, 53-78, (2009) · Zbl 1175.22005
[39] Hu, Z.; Neufang, M.; Ruan, Z-J, Completely bounded multipliers over locally compact quantum groups, Proc. Lond. Math. Soc. (3), 103, 1-39, (2011) · Zbl 1250.22005
[40] Hu, Z.; Neufang, M.; Ruan, Z-J, Module maps over locally compact quantum groups, Stud. Math., 211, 111-145, (2012) · Zbl 1269.22004
[41] Hu, Z.; Neufang, M.; Ruan, Z-J, Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres, Proc. Lond. Math. Soc. (3), 100, 429-458, (2010) · Zbl 1192.43002
[42] Hu, Z.; Neufang, M.; Ruan, Z-J, On topological centre problems and SIN quantum groups, J. Funct. Anal., 257, 610-640, (2009) · Zbl 1184.46047
[43] Junge, M.; Neufang, M.; Ruan, Z-J, A representation theorem for locally compact quantum groups, Int. J. Math., 20, 377-400, (2009) · Zbl 1194.22003
[44] Kalantar, M., Compact operators in regular LCQ groups, Can. Math. Bull., 57, 546-550, (2014) · Zbl 1314.46083
[45] Kalantar, M.: Towards harmonic analysis on locally compact quantum groups from groups to quantum groups and back. Ph.D. thesis, Carleton University (Canada), ProQuest LLC, Ann Arbor, MI (2011)
[46] Kaniuth, E.; Lau, AT-M; Pym, J., On character amenability of Banach algebras, J. Math. Anal. Appl., 344, 942-955, (2008) · Zbl 1151.46035
[47] Kaplanski, I., The structure of certain operator algebras, Trans. Am. Math. Soc., 70, 219-255, (1951)
[48] Kustermans, J., Locally compact quantum groups in the universal setting, Int. J. Math., 12, 289-338, (2001) · Zbl 1111.46311
[49] Kustermans, J.; Vaes, S., Locally compact quantum groups, Ann. Sci. Èole Norm. Sup. (4), 33, 837-934, (2000) · Zbl 1034.46508
[50] Kustermans, J.; Vaes, S., Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand., 92, 68-92, (2003) · Zbl 1034.46067
[51] Lau, AT-M, The second conjugate algebra of the Fourier algebra of a locally compact group, Trans. Am. Math. Soc., 267, 53-63, (1981) · Zbl 0489.43006
[52] Lau, AT-M, Uniformly continuous functionals on the Fourier algebra of any locally compact group, Trans. Am. Math. Soc., 251, 39-59, (1979) · Zbl 0436.43007
[53] Lau, AT-M; Losert, V., On the second conjugate algebra of \(L^1(G)\) of a locally compact group, J. Lond. Math. Soc. (2), 37, 464-470, (1988) · Zbl 0608.43002
[54] Lau, AT-M; Losert, V., The centre of the second conjugate algebra of the Fourier algebra for infinite products of groups, Math. Proc. Camb. Philos. Soc., 138, 27-39, (2005) · Zbl 1067.22004
[55] Lau, AT-M; Ülger, A., Topological centers of certain dual algebras, Trans. Am. Math. Soc., 348, 1191-1212, (1996) · Zbl 0859.43001
[56] Lee, T-K; Wong, T-L, Semiprime algebras with finiteness conditions, Commun. Algebra, 31, 1823-1835, (2003) · Zbl 1045.16008
[57] Losert, V.: The centre of the bidual of Fourier algebras (discrete groups) (preprint) (2002)
[58] Losert, V., Weakly compact multipliers on group algebras, J. Funct. Anal., 213, 466-472, (2004) · Zbl 1069.43001
[59] Mewomo, OT; Maepa, SM, On character amenability of Beurling and second dual algebras, Acta Univ. Apulensis Math. Inform., 38, 67-80, (2014) · Zbl 1340.46038
[60] Monfared, MS, Character amenability of Banach algebras, Math. Proc. Camb. Philos. Soc., 144, 697-706, (2008) · Zbl 1153.46029
[61] Murphy, G.J.: \(C^*\)-Algebras and Operator Theory. Academic Press Inc., Boston (1990) · Zbl 0714.46041
[62] Olubummo, A., Weakly compact B\(^\sharp \)-algebras, Proc. Am. Math. Soc., 14, 905-908, (1963) · Zbl 0119.10801
[63] Ramezanpour, M.; Vishki, HRE, Module homomorphisms and multipliers on locally compact quantum groups, J. Math. Anal. Appl., 359, 581-587, (2009) · Zbl 1181.46036
[64] Ruan, Z-J, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal., 139, 466-499, (1996) · Zbl 0896.46041
[65] Runde, V., Characterizations of compact and discrete quantum groups through second duals, J. Oper. Theory, 60, 415-428, (2008) · Zbl 1164.22001
[66] Runde, V., Uniform continuity over locally compact quantum groups, J. Lond. Math. Soc. (2), 80, 55-71, (2009) · Zbl 1188.46048
[67] Sakai, S., Weakly compact operators on operator algebras, Pac.J. Math., 14, 659-664, (1964) · Zbl 0135.35803
[68] Sołtan, PM, Quantum Bohr compactification, Ill. J. Math., 49, 1245-1270, (2005) · Zbl 1099.46048
[69] Takesaki, M.: Theory of Operator Algebras, I. Springer, New York (1979) · Zbl 0436.46043
[70] Taylor, KF, Geometry of the Fourier algebras and locally compact groups with atomic unitary representations, Math. Ann., 262, 183-190, (1983) · Zbl 0488.43009
[71] Tomiuk, BJ, Arens regularity and the algebra of double multipliers, Proc. Am. Math. Soc., 81, 293-298, (1981) · Zbl 0489.46040
[72] Tomiuk, BJ, Topological algebras with dense socle, J. Funct. Anal., 28, 254-277, (1978) · Zbl 0389.46035
[73] Ülger, A., Arens regularity of weakly sequentially complete Banach algebras, Proc. Am. Math. Soc., 127, 3221-3227, (1999) · Zbl 0930.46046
[74] Vaes, S.: Locally compact quantum groups. Ph.D. Thesis, KU Leuven (2001) · Zbl 1045.46042
[75] Vakilabad, AB; Haghnejad Azar, K.; Jabbari, A., Arens regularity of module actions and weak amenability of Banach algebras, Period. Math. Hung., 71, 224-235, (2015) · Zbl 1363.46036
[76] Daele, A., Locally compact quantum groups. A von Neumann algebra approach, SIGMA Symmetry Integr. Geom. Methods Appl., 10, 41, (2014) · Zbl 1312.46055
[77] Wright, JDM; Ylinen, K., Multilinear maps on products of operator algebras, J. Math. Anal. Appl., 292, 558-570, (2004) · Zbl 1068.47006
[78] Wong, PK, On the Arens product and annihilator algebras, Proc. Am. Math. Soc., 30, 79-83, (1971) · Zbl 0218.46059
[79] Ylinen, K., Weakly completely continuous elements of \(C^*\)-algebras, Proc. Am. Math. Soc., 52, 323-326, (1975) · Zbl 0312.46075
[80] Young, NJ, The irregularity of multiplication in group algebras, Q. J. Math. Oxf. Ser. (2), 24, 59-62, (1973) · Zbl 0252.43009
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.