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On Positive Definiteness Over Locally Compact Quantum Groups

  • Volker Runde (a1) and Ami Viselter (a1) (a2)


The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on “square roots” of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.



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[1] Bédos, E., Murphy, G. J., and Tuset, L., Amenability and coamenability of algebraic quantum groups. Int. J. Math. Math. Sci. 31(2002), no. 10, 577601. http://dx.doi.Org/10.1155/SO16117120210603X
[2] Bédos, E. and Tuset, L., Amenability and co-amenability for locally compact quantum groups. Internat. J. Math. 14(2003), no. 8,865884. http://dx.doi.Org/10.1142/S0129167X03002046
[3] Bergh, J. and Lôfstrôm, J., Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, Berlin, 1976.
[4] Caspers, M., The L?-Fourier transform on locally compact quantum groups J. Operator Theory 69(2013), no. 1, 161193.
[5] Daws, M., Multipliers, self-induced and dual banach algebras. Dissertationes Math. (Rozprawy Mat.) 470(2010), 62 pp.
[6] Daws, M., Completely positive multipliers of quantum groups. Internat. J. Math. 23(2012), no. 12, 1250132, 23 pp. http://dx.doi.Org/10.1142/S0129167X12501327
[7] Daws, M., Kasprzak, P., Skalski, A., and Soltan, P. M., Closed quantum subgroups of locally compact quantum groups. Adv. Math. 231(2012), no. 6, 34733501. http://dx.doi.Org/10.1016/j.aim.2012.09.002
[8] Daws, M. and Salmi, P., Completely positive definite functions and Bochner's theorem for locally compact quantum groups. J. Funct. Anal. 264(2013), no. 7,15251546. http://dx.doi.Org/10.1016/j.jfa.2013.01.017
[9] Desmedt, P., Quaegebeur, J., and Vaes, S., Amenability and the bicrossedproduct construction. Illinois J. Math. 46(2002), no. 4,12591277.
[10] Dixmier, J., C*-algebras. North-Holland Mathematical Library 15, North-Holland Publishing Co., Amsterdam, 1977.
[11] Effros, E. G., Order ideals in a C*-algebra and its dual. Duke Math. J. 30(1963), 391411.http://dx.doi.Org/10.1215/S0012-7094-63-03042-4
[12] Effros, E. G. and Ruan, Z.-J., Discrete quantum groups. I. The Haar measure. Internat. J. Math. 5(1994), no. 5, 681-723. http://dx.doi.Org/10.1142/S0129167X94000358
[13] Enock, M. and Schwartz, J.-M., Kac algebras and duality of locally compact groups. Springer-Verlag, Berlin, 1992.
[14] Eymard, P., L'algèbre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92(1964), 181236.
[15] Forrest, B., Amenability and ideals in A(G). J. Austral. Math. Soc. Ser. A 53(1992), no. 2,143155. http://dx.doi.Org/10.1017/S1446788700035758
[16] Forrest, B. E., Lee, H. H., and Samei, E., Projectivity of modules over Fourier algebras. Proc. Lond.Math. Soc. 102(2011), no. 4, 697730. http://dx.doi.Org/10.1112/plms/pdqO3O
[17] Godement, R., Les fonctions de type positif et la théorie des groupes. Trans. Amer. Math. Soc. 63(1948), no. 1, 184.
[18] Granirer, E. E. and Leinert, M., On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra B(G) and of the measure algebra M(G). Rocky Mountain J. Math. 11(1981), no. 3, 459472.
[19] Haagerup, U., The standard form of von Neumann algebras. Math. Scand. 37(1975), no. 2, 271283.
[20] Haagerup, U., LP-spaces associated with an arbitrary von Neumann algebra. In: Algèbres d'opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), Colloq. Internat.CNRS, vol. 274, CNRS, Paris, 1979, pp. 175184.
[21] Hilsum, M., Les espaces Lp d'une algèbre de von Neumann définies par la dérivée spatiale. J. Funct. Anal. 40(1981), no. 2, 151169.http://dx.doi.Org/10.1016/0022-1236(81)90065-3
[22] Hu, Z., Neufang, M., and Ruan, Z.-J., Module maps over locally compact quantum groups. Studia Math. 211(2012), no. 2, 111145.http://dx.doi.Org/10.4064/sm211-2-2
[23] Izumi, H., Constructions of non-commutative Lp-spaces with a complex parameter arising modular actions. Internat. J. Math. 8(1997), no. 8,10291066. http://dx.doi.Org/10.1142/S0129167X97000494
[24] Izumi, H., Natural bilinear forms, natural sesquilinear forms and the associated duality on non-commutative LP-spaces. Internat. J. Math. 9(1998), no. 8, 9751039.http://dx.doi.Org/10.1142/S0129167X98000439
[25] Jacobs, A., The quantum E(2) group,Ph.D. thesis, Katholieke Universiteit Leuven, 2005, available at
[26] Kaniuth, E. and Lau, A. T., A separation property of positive definite functions on locally compact groups and applications to Fourier algebras. J. Funct. Anal. 175(2000), no. 1, 89110.
[27] Kaniuth, E., On a separation property of positive definite functions on locally compact groups. Math. Z. 243(2003), no. 1, 161177.
[28] Kaniuth, E., Extension and separation properties of positive definite functions on locally compact groups. Trans. Amer. Math. Soc. 359(2007), no. 1, 447463.http://dx.doi.Org/10.1090/S0002-9947-06-03969-9
[29] Kustermans, J., Locally compact quantum groups in the universal setting. Internat. J. Math. 12(2001), no. 3, 289338.
[30] Kustermans, J. and Vaes, S., Locally compact quantum groups. Ann. Sci. École Norm. Sup. 33(2000), no. 6, 837934.
[31] Kustermans, J., Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92(2003), no. 1, 6892.
[32] Kyed, D., A cohomological description of property (T)for quantum groups. J. Funct. Anal. 261(2011), no. 6, 14691493. http://dx.doi.Org/10.1016/j.jfa.2O11.05.010
[33] Lance, E. C., Hilbert C* -modules. A toolkit for operator algebraists. London Mathematical Society Lecture Note Series, 210, Cambridge University Press, Cambridge, 1995.
[34] Lau, A. T.-M. and Losert, V., Weak* -closed complemented invariant subspaces of L∞ (G) and amenable locally compact groups. Pacific J. Math. 123(1986), no. 1,149159. http://dx.doi.Org/10.2140/pjm.1986.123.149
[35 Masuda, T., Nakagami, Y., and Woronowicz, S. L., A C* -algebraic framework for quantum groups. Internat. J. Math. 14(2003), no. 9, 9031001.http://dx.doi.Org/10.1142/S0129167X03002071
[36] Meyer, R., Roy, S., and Woronowicz, S. L., Homomorphisms of quantum groups. Munster J. Math.5(2012), 124.
[37] Packer, J. A. and Raeburn, I., Twisted crossed products of C*-algebras. Math. Proc. Cambridge Philos. Soc. 106(1989), no. 2, 293311.
[38] Perdrizet, F., Éléments positifs relatifs à une algèbre hilbertienne à gauche. Compositio Math. 23(1971), 2547.
[39] Phillips, J., Positive integrable elements relative to a left Hilbert algebra. J. Funct. Anal. 13(1973), 390409. http://dx.doi.Org/10.1016/0022-1236(73)90057-8
[40] Raïkov, D.A, On various types of convergence of positive definite functions. Doklady Akad. Nauk SSSR (N.S.) 58(1947), 12791282.
[41] Runde, V., Characterizations of compact and discrete quantum groups through second duals. J. Operator Theory 60(2008), no. 2,415428.
[42] Runde, V., Uniform continuity over locally compact quantum groups. J. Lond. Math. Soc. 80(2009), no. 1, 5571.http://dx.doi.Org/10.1112/jlms/jdpO11
[43] Ç.Strâtilâ, , Modular theory in operator algebras. Abacus Press, Tunbridge Wells, England, 1981.
[44] Takesaki, M., Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes in Mathematics 128, Springer-Verlag, Berlin, 1970.
[45] Takesaki, M., Theory of operator algebras. I. Encyclopaedia of Mathematical Sciences, vol. 124, Springer-Verlag, Berlin, 2002.
[46] Takesaki, M., Theory of operator algebras. II. Encyclopaedia of Mathematical Sciences,vol. 125, Springer-Verlag, Berlin,2003.
[47] Takesaki, M. and Tatsuuma, N., Duality and subgroups, II. J. Funct. Anal. 11(1972), no. 2,184190. http://dx.doi.Org/1 0.1016/0022-1236(72)90087-0
[48] Terp, M., Lpspaces associated with von Neumann algebras,Notes, Report No. 3a+3b, Kobenhavns Universitets Matematiske Institut, June 1981.
[49] Terp, M., Interpolation spaces between a von Neumann algebra and its predual. J. Operator Theory 8(1982), no. 2, 327360.
[50] Vaes, S. and Vainerman, L., Extensions of locally compact quantum groups and the bicrossed product construction. Adv. Math. 175(2003), no. 1,1101. http://dx.doi.Org/10.101 6/S0001 -8708(02)00040-3
[51] Vaes, S., On low-dimensional locally compact quantum groups. In: Locally compact quantum groups and groupoids (Strasbourg, 2002), IRMA Lect. Math. Theor. Phys., vol. 2, de Gruyter, Berlin, 2003, pp. 127187.
[52] Vaes, S. and Vergnioux, R., The boundary of universal discrete quantum groups, exactness, and factoriality. Duke Math. J. 140(2007), no. 1, 3584. http://dx.doi.Org/10.1215/S001 2-7094-07-14012-2
[53] Valette, A., On Godement's characterisation of amenability. Bull. Austral. Math. Soc. 57(1998), no. 1, 153158.
[54] Van Daele, A., Discrete quantum groups. J. Algebra 180(1996), no. 2, 431444.
[55] Van Daele, A., Locally compact quantum groups. A von Neumann algebra approach. SIGMA Symmetry Integrability Geom. Methods Appl. 10(2014), Paper 082, 41 pp.
[56] Woronowicz, S. L., Compact quantum groups. In: Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 845884.
[57] Yoshizawa, H., On some types of convergence of positive definite functions. Osaka Math. J. 1(1949), no. 1, 9094.
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