×

Mapping ideals of quantum group multipliers. (English) Zbl 1464.46071

Summary: We study the dual relationship between quantum group convolution maps \(L^1 (\mathbb{G}) \to L^\infty (\mathbb{G})\) and completely bounded multipliers of \(\hat{\mathbb{G}}\). For a large class of locally compact quantum groups \(\mathbb{G}\), we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with \(M_{c b} (L^1(\hat{\mathbb{G}}))\), yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter coinciding with \(\ell^1 (\hat{b \mathbb{G}})\), where \(b \mathbb{G}\) is the quantum Bohr compactification of \(\mathbb{G}\). For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with \(C (b \mathbb{G})\). Our techniques comprise a mixture of operator space theory and abstract harmonic analysis, including Fubini tensor products, the non-commutative Grothendieck inequality, quantum Eberlein compactifications, and a suitable notion of inner co-amenable quantum group, that we introduce and for which we exhibit examples from the bicrossed product construction. Our main results are new even in the setting of group von Neumann algebras \(VN(G)\) for quasi-SIN locally compact groups \(G\).

MSC:

46L67 Quantum groups (operator algebraic aspects)
46L07 Operator spaces and completely bounded maps
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46M05 Tensor products in functional analysis
47L25 Operator spaces (= matricially normed spaces)
22D35 Duality theorems for locally compact groups
22D15 Group algebras of locally compact groups
43A10 Measure algebras on groups, semigroups, etc.
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alaghmandan, M.; Todorov, I. G.; Turowska, L., Completely bounded bimodule maps and spectral synthesis, Int. J. Math., 28, 10, Article 1750067 pp. (2017) · Zbl 1385.46038
[2] Aristov, O. Y.; Runde, V.; Spronk, N., Operator biflatness of the Fourier algebra and approximate indicators for subgroups, J. Funct. Anal., 209, 2, 367-387 (2004) · Zbl 1052.22005
[3] Banica, T.; Chirvasitu, A., Thoma type results for discrete quantum groups, Int. J. Math., 28, 14, Article 1750103 pp. (2017) · Zbl 1458.46057
[4] Bierstedt, K. D., Introduction to Topological Tensor Products, Paderborn University Lecture Notes (2009), Mathematical Institute, Paderborn University, Typeset by S.-A. Wegner
[5] Blackadar, B., Operator Algebras: Theory of \(C^\ast \)-Algebras and Von Neumann Algebras, Encyclopaedia of Mathematical Sciences, vol. 122 (2006), Springer-Verlag Berlin Heidelberg · Zbl 1092.46003
[6] Blecher, D. P., A completely bounded characterization of operator algebras, Math. Ann., 303, 2, 227-239 (1995) · Zbl 0892.47048
[7] Blecher, D. P.; Le Merdy, C., Operator Algebras and Their Modules: An Operator Space Approach, London Mathematical Society Monographs, New Series, vol. 30 (2004), Clarendon Press: Clarendon Press Oxford · Zbl 1061.47002
[8] Blecher, D. P.; Smith, R. R., The dual of the Haagerup tensor product, J. Lond. Math. Soc. (2), 45, 1, 126-144 (1992) · Zbl 0712.46029
[9] Bojeko, M.; Fendler, G., Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Unione Mat. Ital., A (6), 3, 2, 297-302 (1984) · Zbl 0564.43004
[10] Brannan, M.; Youn, S.-G., On the similarity problem for locally compact quantum groups, J. Funct. Anal., 276, 4, 1313-1337 (2019) · Zbl 1430.22007
[11] Caspers, M.; Lee, H. H.; Ricard, E., Operator biflatness of the \(L^1\)-algebras of compact quantum groups, J. Reine Angew. Math., 700, 235-244 (2015) · Zbl 1322.46036
[12] Cowling, M.; Haagerup, U., Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math., 96, 3, 507-549 (1989) · Zbl 0681.43012
[13] Crann, J., Inner amenability and approximation properties of locally compact quantum groups, Indiana Univ. Math. J., 68, 6, 1721-1766 (2019) · Zbl 1464.46072
[14] Crann, J.; Tanko, Z., On the operator homology of the Fourier algebra and its cb-multiplier completion, J. Funct. Anal., 273, 7, 2521-2545 (2017) · Zbl 1384.46035
[15] Crombez, G.; Govaerts, W., Nuclear multipliers from \(L_1(G)\) into \(L_p(G)\), Simon Stevin, 60, 4, 347-351 (1986) · Zbl 0637.43004
[16] Dales, H. G.; Lau, A. T.-M., The second duals of Beurling algebras, Mem. Am. Math. Soc., 177, 836 (2005) · Zbl 1075.43003
[17] Das, B.; Daws, M., Quantum Eberlein compactifications and invariant means, Indiana Univ. Math. J., 65, 1, 307-352 (2016) · Zbl 1354.43001
[18] Das, B.; Daws, M.; Salmi, P., Admissibility conjecture and Kazhdan’s property (T) for quantum groups, J. Funct. Anal., 276, 11, 3484-3510 (2019) · Zbl 1412.22014
[19] Davis, W. J.; Figiel, T.; Johnson, W. B.; Pelczyński, A., Factoring weakly compact operators, J. Funct. Anal., 17, 311-327 (1974) · Zbl 0306.46020
[20] Daws, M., Multipliers of locally compact quantum groups via Hilbert \(C^\ast \)-modules, J. Lond. Math. Soc. (2), 84, 2, 385-407 (2011) · Zbl 1235.43004
[21] Daws, M., Completely positive multipliers of quantum groups, Int. J. Math., 23, 12, 125-132 (2012) · Zbl 1282.43002
[22] Daws, M., Remarks on the quantum Bohr compactification, Ill. J. Math., 57, 4, 1131-1171 (2013) · Zbl 1305.43006
[23] Daws, M.; Skalski, A.; Viselter, A., Around property (T) for quantum groups, Commun. Math. Phys., 353, 1, 69-118 (2017) · Zbl 1371.46060
[24] Dunkl, C. F.; Ramirez, D. E., Weakly almost periodic functionals on the Fourier algebra, Trans. Am. Math. Soc., 185, 501-514 (1973) · Zbl 0271.43009
[25] Effros, E. G.; Junge, M.; Ruan, Z.-J., Integral mappings and the principle of local reflexivity for noncommutative \(L^1\)-spaces, Ann. Math. (2), 151, 1, 59-92 (2000) · Zbl 0957.47051
[26] Effros, E. G.; Ruan, Z.-J., The Grothendieck-Pietsch and Dvoretzky-Rogers theorems for operator spaces, J. Funct. Anal., 122, 2, 428-450 (1994) · Zbl 0802.46014
[27] Effros, E. G.; Ruan, Z.-J., Operator Spaces, London Mathematical Society Monographs. New Series, vol. 23 (2000), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York · Zbl 0969.46002
[28] Fima, P.; Mukherjee, K.; Patri, I., On compact bicrossed products, J. Noncommut. Geom., 11, 4, 1521-1591 (2017) · Zbl 1410.46054
[29] Forrest, B. E., Arens regularity and discrete groups, Pac. J. Math., 151, 2, 217-227 (1991) · Zbl 0746.43002
[30] Gilbert, J. E., \( L^p\)-convolution operators and tensor products of Banach spaces, Bull. Am. Math. Soc., 80, 1127-1132 (1974) · Zbl 0324.43003
[31] J.E. Gilbert, \( L^p\)-convolution operators and tensor products of Banach spaces I, II, III, preprints, 1973-1974.
[32] U. Haagerup, Decomposition of completely bounded maps on operator algebras, 1980, Unpublished manuscript.
[33] Haagerup, U.; de Laat, T., Simple Lie groups without the approximation property, Duke Math. J., 162, 5, 925-964 (2013) · Zbl 1266.22008
[34] Haagerup, U.; Kraus, J., Approximation properties for group \(C^\ast \)-algebras and group von Neumann algebras, Trans. Am. Math. Soc., 44, 2, 667-699 (1994) · Zbl 0806.43002
[35] Haagerup, U.; Musat, M., The Effros-Ruan conjecture for bilinear forms on \(C^\ast \)-algebras, Invent. Math., 174, 1, 139-163 (2008) · Zbl 1188.46034
[36] Hu, Z.; Neufang, M.; Ruan, Z.-J., On topological centre problems and SIN quantum groups, J. Funct. Anal., 257, 2, 610-640 (2009) · Zbl 1184.46047
[37] Hu, Z.; Neufang, M.; Ruan, Z.-J., Completely bounded multipliers over locally compact quantum groups, Proc. Lond. Math. Soc. (3), 103, 1, 1-39 (2011) · Zbl 1250.22005
[38] Hu, Z.; Neufang, M.; Ruan, Z.-J., Module maps over locally compact quantum groups, Stud. Math., 211, 2, 111-145 (2012) · Zbl 1269.22004
[39] Jolissaint, P., A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math., 63, 2, 311-313 (1992) · Zbl 0774.43003
[40] Junge, M.; Neufang, M.; Ruan, Z.-J., A representation theorem for locally compact quantum groups, Int. J. Math., 20, 3, 377-400 (2009) · Zbl 1194.22003
[41] Kasprzak, P.; Sołtan, P. M., Embeddable quantum homogeneous spaces, J. Math. Anal. Appl., 411, 2, 574-591 (2014) · Zbl 1337.46047
[42] Knudby, S., The weak Haagerup property, Trans. Am. Math. Soc., 368, 5, 3469-3508 (2016) · Zbl 1332.22008
[43] Krajczok, J.; Sołtan, P. M., Compact quantum groups with representations of bounded degree, J. Oper. Theory, 80, 2, 415-428 (2018) · Zbl 1474.46119
[44] Kustermans, J., Locally compact quantum groups in the universal setting, Int. J. Math., 12, 3, 289-338 (2001) · Zbl 1111.46311
[45] Kustermans, J.; Vaes, S., Locally compact quantum groups, Ann. Sci. Éc. Norm. Supér. (4), 33, 6, 837-934 (2000) · Zbl 1034.46508
[46] Kustermans, J.; Vaes, S., Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand., 92, 1, 68-92 (2003) · Zbl 1034.46067
[47] Lance, E. C., Hilbert \(C^\ast \)-Modules, London Mathematical Society Lecture Note Series, vol. 210 (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0822.46080
[48] Lee, H. H.; Samei, E.; Spronk, N., Similarity degree of Fourier algebras, J. Funct. Anal., 271, 3, 593-609 (2016) · Zbl 1354.46050
[49] Lee, H. H.; Samei, E.; Spronk, N., Corrigendum: similarity degree of Fourier algebras, J. Funct. Anal., 277, 3, 958-964 (2019) · Zbl 1441.46037
[50] Losert, V.; Rindler, H., Asymptotically central functions and invariant extensions of Dirac measure, (Probability Measures on Groups, VII. Probability Measures on Groups, VII, Oberwolfach, 1983. Probability Measures on Groups, VII. Probability Measures on Groups, VII, Oberwolfach, 1983, Lecture Notes in Math., vol. 1064 (1984), Springer: Springer Berlin), 368-378 · Zbl 0573.43003
[51] Majid, S., Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts, and the classical Yang-Baxter equations, J. Funct. Anal., 95, 291-319 (1991) · Zbl 0741.46033
[52] Moore, C. C., Groups with finite dimensional irreducible representations, Trans. Am. Math. Soc., 166, 401-410 (1972) · Zbl 0236.22010
[53] Neufang, M.; Ruan, Z.-J.; Spronk, N., Completely isometric representations of \(M_{c b} A(G)\) and \(U C B ( \hat{G} )^\ast \), Trans. Am. Math. Soc., 360, 3, 1133-1161 (2008) · Zbl 1142.22002
[54] Pfitzner, H.; Schlüchtermann, G., Factorization of completely bounded weakly compact operators (1997), preprint
[55] Pisier, G., Grothendieck’s theorem, past and present, Bull. Am. Math. Soc. (N.S.), 49, 2, 237-323 (2012) · Zbl 1244.46006
[56] Pisier, G.; Shlyakhtenko, D., Grothendieck’s theorem for operator spaces, Invent. Math., 150, 1, 185-217 (2002) · Zbl 1033.46044
[57] Quigg, J. C., Approximately periodic functionals on \(C^\ast \)-algebras and von Neumann algebras, Can. J. Math., 37, 5, 769-784 (1985)
[58] Racher, G., The nuclear multipliers from \(L^1(G)\) into \(L^\infty(G)\), J. Funct. Anal., 122, 2, 279-306 (1994) · Zbl 0845.43003
[59] Ruan, Z.-J., Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal., 139, 2, 466-499 (1996) · Zbl 0896.46041
[60] Ruan, Z.-J.; Xu, G., Splitting properties of operator bimodules and operator amenability of Kac algebras, (Operator Theory, Operator Algebras and Related Topics. Operator Theory, Operator Algebras and Related Topics, Timisoara, 1996 (1997), Theta Found.: Theta Found. Bucharest), 193-216 · Zbl 0942.46033
[61] Runde, V.; Spronk, N., Operator amenability of Fourier-Stieltjes algebras, Math. Proc. Camb. Philos. Soc., 136, 3, 675-686 (2004) · Zbl 1052.43003
[62] Runde, V., Uniform continuity over locally compact quantum groups, J. Lond. Math. Soc. (2), 80, 1, 55-71 (2009) · Zbl 1188.46048
[63] Runde, V., Completely almost periodic functionals, Arch. Math. (Basel), 97, 4, 325-331 (2011) · Zbl 1250.47083
[64] Saar, H., Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen \(C^\ast \)-Algebra (1982), Saarland University, Diploma Thesis
[65] Sakai, S., Weakly compact operators on operator algebras, Pac. J. Math., 14, 659-664 (1964) · Zbl 0135.35803
[66] Smith, R. R., Completely bounded module maps and the Haagerup tensor product, J. Funct. Anal., 102, 156-175 (1991) · Zbl 0745.46060
[67] Sołtan, P. M., Quantum Bohr compactification, Ill. J. Math., 49, 4, 1245-1270 (2005) · Zbl 1099.46048
[68] Spronk, N., Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras, Proc. Lond. Math. Soc. (3), 89, 1, 161-192 (2004) · Zbl 1047.43008
[69] Stokke, R., Quasi-central bounded approximate identities in group algebras of locally compact groups, Ill. J. Math., 48, 1, 151-170 (2004) · Zbl 1037.43004
[70] Takesaki, M., A characterization of group algebras as a converse of Tannaka-Stinespring-Tatsuuma duality theorem, Am. J. Math., 91, 529-564 (1969) · Zbl 0182.18103
[71] Takesaki, M., Theory of Operator Algebras II, Encyclopedia of Mathematical Sciences, vol. 125 (2003), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 1059.46031
[72] Takesaki, M.; Tatsuuma, N., Duality and subgroups II, J. Funct. Anal., 11, 184-190 (1972) · Zbl 0245.46090
[73] Vaes, S., The unitary implementation of a locally compact quantum group action, J. Funct. Anal., 180, 2, 426-480 (2001) · Zbl 1011.46058
[74] Vaes, S.; Vainerman, L., Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math., 175, 1, 1-101 (2003) · Zbl 1034.46068
[75] Van Daele, A., Locally compact quantum groups. A von Neumann algebra approach, SIGMA, 10, Article 082 pp. (2014) · Zbl 1312.46055
[76] Wilson, B., A Hilbert space approach to approximate diagonals for locally compact quantum groups, Banach J. Math. Anal., 9, 3, 248-260 (2015) · Zbl 1311.43004
[77] Woronowicz, S. L., Compact quantum groups, (Symétries Quantiques. Symétries Quantiques, Les Houches, 1995 (1998), North-Holland: North-Holland Amsterdam), 845-884 · Zbl 0997.46045
[78] Xu, G., Herz-Schur multipliers and weakly almost periodic functions on locally compact groups, Trans. Am. Math. Soc., 349, 6, 2525-2536 (1997) · Zbl 0874.43004
[79] Youn, S.-G., A theorem for random Fourier series on compact quantum groups, Indiana Univ. Math. J., 69, 3, 887-910 (2020), in press · Zbl 1475.46061
[80] S.-G. Youn, Private communication.
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.