Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers

Nemati, Mehdi ; Rizi, Maryam Rajaei

Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 825-836

Résumé

Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.

Détail

DOI : 10.4153/S0008439520000077

Mots-clés : amenability, Arens regularity, locally compact quantum group, (weakly) compact multiplier, module homomorphism

Bibliographie

[1] Amini, M., Kalantar, M., Medghalchi, A., Mollakhalili, A., and Neufang, M., Compact elements and operators of quantum groups. Glasg. Math. J. 59(2017), 445–462. https://doi.org/10.1017/S0017089516000276 Google Scholar | DOI

[2] Bédos, E. and Tuset, L., Amenability and co-amenability for locally compact quantum groups. Internat. J. Math. 14(2003), 865–884. https://doi.org/10.1142/S0129167X03002046 Google Scholar | DOI

[3] Daws, M. and Lepham, H., Isometries between quantum convolution algebras. Q. J. Math. 64(2013), 373–396. https://doi.org/10.1093/qmath/has008 Google Scholar | DOI

[4] Duncan, J. and Hosseiniun, S. A. R., The second dual of a Banach algebra. Proc. Roy. Soc. Edinburgh Sect. A 84(1979), 309–325. https://doi.org/10.1017/S0308210500017170 Google Scholar | DOI

[5] Forrest, B., Arens regularity and discrete groups. Pacific J. Math. 151(1991), 217–227. Google Scholar | DOI

[6] Ghahramani, F. and Lau, A. T.-M., Multipliers and ideals in second conjugate algebras related to locally compact groups. J. Funct. Anal. 132(1995), 170–191. https://doi.org/10.1006/jfan.1995.1104 Google Scholar | DOI

[7] Ghahramani, F. and Lau, A. T.-M., Multipliers and modulus on Banach algebras related to locally compact groups. J. Funct. Anal. 150(1997), 478–497. https://doi.org/10.1006/jfan.1997.3133 Google Scholar | DOI

[8] Ghanei, M., Nasr-Isfahani, R., and Nemati, M., A homological property and Arens regularity of locally compact quantum groups. Canad. Math. Bull. 60(2017), no. 1, 122–130. https://doi.org/10.4153/CMB-2016-052-x Google Scholar | DOI

[9] Hu, Z., Neufang, M., and Ruan, Z.-J., On topological center problems and SIN quantum groups. J. Funct. Anal. 257(2009), 610–640. https://doi.org/10.1016/j.jfa.2009.02.004 Google Scholar | DOI

[10] Kalantar, M. and Neufang, M., From quantum groups to groups. Canadian J. Math. 65(2013), 1073–1094. https://doi.org/10.4153/CJM-2012-047-x Google Scholar | DOI

[11] Kaniuth, E., Lau, A. T.-M., and Pym, J., On 𝜑-amenability of Banach algebras. Math. Proc. Cambridge Philos. Soc. 144(2008), 85–96. https://doi.org/10.1017/S0305004107000874 Google Scholar | DOI

[12] Kaniuth, E., Lau, A. T.-M., and Pym, J., On character amenability of Banach algebras. J. Math. Anal. Appl. 344(2008), 942–955. https://doi.org/10.1016/j.jmaa.2008.03.037 Google Scholar | DOI

[13] Kustermans, J. and Vaes, S., Locally compact quantum groups. Ann. Sci. École Norm. Sup. 33(2000), 837–934. https://doi.org/10.1016/S0012-9593(00)01055-7 Google Scholar | DOI

[14] Kustermans, J. and Vaes, S., Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92(2003), 68–92. https://doi.org/10.7146/math.scand.a-14394 Google Scholar | DOI

[15] Lau, A. T.-M., Uniformly continuous functionals on the Fourier algebra of any locally compact group. Trans. Am. Math. Soc. 251(1979), 39–59. https://doi.org/10.2307/1998682 Google Scholar | DOI

[16] Losert, V., Weakly compact multipliers on group algebras. J. Funct. Anal. 213(2004), 466–472. https://doi.org/10.1016/j.jfa.2003.10.012 Google Scholar | DOI

[17] Ruan, Z.-J., Amenability of Hopf von Neumann algebras and Kac algebras. J. Funct. Anal. 139(1996), 466–499. https://doi.org/10.1006/jfan.1996.0093 Google Scholar | DOI

[18] Runde, V., Characterizations of compact and discrete quantum groups through second duals. J. Operator Theory 60(2008), 415–428. Google Scholar

[19] Sakai, S., Weakly compact operators on operator algebras. Pac. J. Math. 14(1964), 659–664. Google Scholar | DOI

[20] Ulger, A., Arens regularity sometimes implies RNP. Pacific J. Math. 143(1990), 377–399. Google Scholar | DOI