# zbMATH — the first resource for mathematics

On topological centre problems and SIN quantum groups. (English) Zbl 1184.46047
Let $$A$$ be a Banach algebra with faithful multiplication. There are two ways to extend the multiplication of $$A$$ to its second dual: the left Arens product $$\square$$ and the right Arens product $$\lozenge$$. The left Arens product is defined via actions
$\langle fa,b\rangle = \langle f, ab\rangle,\qquad \langle n\square f,a\rangle = \langle n, fa\rangle,\qquad \langle m\square n,f\rangle = \langle m, n\square f\rangle,$ where $$a,b\in A$$, $$f\in A^*$$ and $$m,n\in A^{**}$$. The right Arens product and the related actions are defined analogously. Denote the closed linear span of $$A^* A$$ by $$\langle A^* A\rangle$$. Then $$\langle A^* A\rangle^*$$ is a quotient Banach algebra of $$(A^{**},\square)$$. Now all right translations in $$(A^{**},\square)$$ are weak* continuous; the collection of all elements $$m$$ in $$A^{**}$$ such that the left translation by $$m$$ is weak* continuous is called the topological centre of $$(A^{**},\square)$$ and is denoted by $${\mathcal Z}_t(A^{**},\square)$$. The topological centre $${\mathcal Z}_t(\langle A^*A\rangle^*)$$ of $$\langle A^* A\rangle^*$$ is defined similarly.
The authors define
$\langle A^* A\rangle^*_R = \{m \in \langle A^* A\rangle^*: \langle A^* A\rangle^* \lozenge m\subseteq \langle A^* A\rangle^*\}$ and equip $$\langle A^* A\rangle^*_R$$ with a new product using the right Arens product of $$A^{**}$$. With the help of $$\langle A^* A\rangle^*_R$$, the authors give a new characterisation for the topological centre of $$\langle A^* A\rangle^*$$: ${\mathcal Z}_t(\langle A^*A\rangle^*) = \{ m\in \langle A^* A\rangle^*_R: m\square n = m\lozenge n \quad \text{for every } n\in \langle A^* A\rangle^*\},$ which compares with the well-known fact that ${\mathcal Z}_t(A^{**},\square) = \{ m\in A^{**}: m\square n = m\lozenge n \quad \text{for every } n\in A^{**}\}.$ As a corollary, the authors extend a characterisation of the topological centre of $$\langle A^* A\rangle^*$$ due to A. T.-M. Lau and A. Ülger [Trans. Am. Math. Soc. 348, 1191–1212 (1996; Zbl 0859.43001)] by removing the hypothesis of the existence of a bounded approximate identity in $$A$$. The characterisation says that $$m$$ in $$\langle A^* A\rangle^*$$ is in $${\mathcal Z}_t(\langle A^*A\rangle^*)$$ if and only if $$A\cdot m\subseteq {\mathcal Z}_t(A^{**},\square)$$. Other related results are proved.
Now consider a locally compact quantum group $${\mathbb G}$$. Then $$L^1({\mathbb G})$$, the predual of the von Neumann algebra $$L^\infty({\mathbb G})$$ associated with $${\mathbb G}$$, is a Banach algebra with faithful multiplication, so the machinery developed in the paper is applicable to $$A = L^1({\mathbb G})$$. Write $$LUC({\mathbb G}) = \langle L^\infty({\mathbb G})L^1({\mathbb G})\rangle$$ and $$RUC({\mathbb G}) = \langle L^1({\mathbb G})L^\infty({\mathbb G})\rangle$$. A locally compact quantum group $${\mathbb G}$$ is said to be SIN if $$LUC({\mathbb G}) = RUC({\mathbb G})$$ (which is equivalent, in the case when $${\mathbb G}$$ is a usual locally compact group, with the existence of a neighbourhood base at the identity consisting of invariant neighbourhoods). The authors show, for example, that $${\mathbb G}$$ is a co-amenable SIN quantum group if and only if the Banach algebra $$LUC({\mathbb G})^*_R$$ is unital. Other equivalent conditions are also given. It is worth to note that the conditions equivalent with the SIN property are new even for locally compact groups.
A Banach algebra $$A$$ is said to be left strongly Arens irregular if $${\mathcal Z}_t(A^{**},\square) = A$$ and left quotient strongly Arens irregular if $${\mathcal Z}_t(\langle A^*A\rangle^*)$$ is contained in the opposite right multiplier algebra $$RM(A)$$. The authors study how the left strong Arens irregularity of $$A$$ is related to the left quotient strong Arens irregularity. Many of the results concern the class of Banach algebras introduced by the authors in [Proc. Lond. Math. Soc. (3) 100, No. 2, 429–458 (2010; Zbl 1192.43002)]. Finally, the authors consider several examples and thereby resolve some open questions asked by A. T.-M. Lau and A. Ülger [op. cit].
One of the conclusions of the paper is that the SIN property is intrinsically related to topological centre problems. The paper also exemplifies the principle that studying more general objects, such as locally compact quantum groups instead of locally compact groups, can shed light on old issues, as is the case with SIN groups in this paper.
Reviewer: Pekka Salmi (Oulu)

##### MSC:
 46H05 General theory of topological algebras 43A20 $$L^1$$-algebras on groups, semigroups, etc. 20G42 Quantum groups (quantized function algebras) and their representations
Full Text:
##### References:
  Arens, R., The adjoint of a bilinear operation, Proc. amer. math. soc., 2, 839-848, (1951) · Zbl 0044.32601  Baker, J.; Lau, A.T.-M.; Pym, J., Module homomorphisms and topological centres associated with weakly sequentially complete Banach algebras, J. funct. anal., 158, 186-208, (1998) · Zbl 0911.46030  Bédos, E.; Tuset, L., Amenability and co-amenability for locally compact quantum groups, Internat. J. math., 14, 865-884, (2003) · Zbl 1051.46047  Berglund, J.F.; Junghenn, H.D.; Milnes, P., Analysis on semigroups. function spaces, compactifications, representations, Canad. math. soc. ser. monogr. adv. texts, (1989), Wiley-Interscience Publ. John Wiley & Sons, Inc. New York · Zbl 0727.22001  Dales, H.G., Banach algebras and automatic continuity, London math. soc. monogr. new ser., vol. 24, (2000), Oxford University Press New York · Zbl 0981.46043  Dales, H.G.; Lau, A.T.-M., The second duals of Beurling algebras, Mem. amer. math. soc., 177, (2005), no. 836 · Zbl 1075.43003  H.G. Dales, A.T.-M. Lau, D. Strauss, Banach algebras on semigroups and their compactifications, Mem. Amer. Math. Soc., in press · Zbl 1192.43001  De Cannière, J.; Haagerup, U., Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. math., 107, 455-500, (1985) · Zbl 0577.43002  Farhadi, H.; Ghahramani, F., Involutions on the second duals of group algebras and a multiplier problem, Proc. edinb. math. soc., 50, 153-161, (2007) · Zbl 1121.43001  Forrest, B., Arens regularity and discrete groups, Pacific J. math., 151, 217-227, (1991) · Zbl 0746.43002  Ghahramani, F.; McClure, J.P.; Meng, M., On asymmetry of topological centres of the second duals of Banach algebras, Proc. amer. math. soc., 126, 1765-1768, (1998) · Zbl 0894.46034  Grosser, M., Bidualräume und vervollständigungen von banachmoduln, Lecture notes in math., vol. 717, (1979), Springer Berlin · Zbl 0412.46005  Grosser, M.; Losert, V., The norm-strict bidual of a Banach algebra and the dual of $$C_u(G)$$, Manuscripta math., 45, 127-146, (1984) · Zbl 0527.46037  Haagerup, U.; Kraus, J., Approximation properties for group $$C^\ast$$-algebras and group von Neumann algebras, Trans. amer. math. soc., 344, 667-699, (1994) · Zbl 0806.43002  Hewitt, E.; Ross, K.A., Abstract harmonic analysis I, (1979), Springer-Verlag New York · Zbl 0837.43002  Z. Hu, M. Neufang, Z.-J. Ruan, Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres, preprint, 2007 · Zbl 1192.43002  Z. Hu, M. Neufang, Z.-J. Ruan, Completely bounded multipliers on co-amenable locally compact quantum groups, in preparation · Zbl 1250.22005  M. Junge, M. Neufang, Z.-J. Ruan, A representation for locally compact quantum groups, Internat. J. Math., in press · Zbl 1194.22003  Kaniuth, E.; Lau, A.T.-M., A separation property of positive definite functions on locally compact groups and applications to Fourier algebras, J. funct. anal., 175, 89-110, (2000) · Zbl 0953.43002  Kraus, J.; Ruan, Z.-J., Multipliers of Kac algebras, Internat. J. math., 8, 213-248, (1996) · Zbl 0870.46037  Kustermans, J.; Vaes, S., Locally compact quantum groups, Ann. sci. école norm. sup., 33, 837-934, (2000) · Zbl 1034.46508  Kustermans, J.; Vaes, S., Locally compact quantum groups in the von Neumann algebraic setting, Math. scand., 92, 68-92, (2003) · Zbl 1034.46067  Lau, A.T.-M., Operators which commute with convolutions on subspaces of $$L_\infty(G)$$, Colloq. math., 39, 351-359, (1978) · Zbl 0411.47025  Lau, A.T.-M., Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups, Fund. math., 118, 161-175, (1983) · Zbl 0545.46051  Lau, A.T.-M., Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. proc. Cambridge philos. soc., 99, 273-283, (1986) · Zbl 0591.43003  Lau, A.T.-M.; Losert, V., On the second conjugate algebra of $$L_1(G)$$ of a locally compact group, J. London math. soc. (2), 37, 464-470, (1988) · Zbl 0608.43002  Lau, A.T.-M.; Losert, V., The $$C^\ast$$-algebra generated by operators with compact support on a locally compact group, J. funct. anal., 112, 1-30, (1993) · Zbl 0788.22006  Lau, A.T.-M.; Ülger, A., Topological centers of certain dual algebras, Trans. amer. math. soc., 348, 1191-1212, (1996) · Zbl 0859.43001  Leptin, H., Sur l’algèbre de Fourier d’un groupe localement compact, C. R. acad. sci. Paris Sér. A, 266, 1180-1182, (1968) · Zbl 0169.46501  Losert, V., Some properties of groups without the property $$P_1$$, Comment. math. helv., 54, 133-139, (1979) · Zbl 0396.43007  Milnes, P., Uniformity and uniformly continuous functions for locally compact groups, Proc. amer. math. soc., 109, 567-570, (1990) · Zbl 0697.22005  Mitchell, T., Topological semigroups and fixed points, Illinois J. math., 14, 630-641, (1970) · Zbl 0219.22003  Mosak, R.D., Central functions in group algebras, Proc. amer. math. soc., 29, 613-616, (1971) · Zbl 0204.35301  M. Neufang, Abstrakte Harmonische Analyse und Modulhomomorphismen über von Neumann-Algebren, PhD thesis at University of Saarland, Saarbrücken, Germany, 2000  Palmer, T.W., Banach algebras and general theory of ∗-algebras, vol. 1, (1994), Cambridge University Press Cambridge  Ramamohana Rao, C., Invariant means on spaces of continuous or measurable functions, Trans. amer. math. soc., 114, 187-196, (1965) · Zbl 0139.30901  Ruan, Z.-J., The operator amenability of $$A(G)$$, Amer. J. math., 117, 1449-1474, (1995) · Zbl 0842.43004  Ruan, Z.-J., Amenability of Hopf von Neumann algebras and Kac algebras, J. funct. anal., 139, 466-499, (1996) · Zbl 0896.46041  Rudin, W., Homomorphisms and translations in $$L_\infty(G)$$, Adv. math., 16, 72-90, (1975) · Zbl 0297.22009  Runde, V., Characterizations of compact and discrete quantum groups through second duals, J. operator theory, 60, 415-428, (2008) · Zbl 1164.22001  V. Runde, Uniform continuity over locally compact quantum groups, preprint, 2008  Takesaki, M., Theory of operator algebras I, operator algebras and non-commutative geometry V, Encyclopaedia math. sci., vol. 124, (2002), Springer-Verlag Berlin · Zbl 0990.46034  Talagrand, M., Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations, Ann. inst. Fourier (Grenoble), 32, 39-69, (1982) · Zbl 0452.28004  Tomatsu, R., Amenable discrete quantum groups, J. math. soc. Japan, 58, 949-964, (2006) · Zbl 1129.46061  A. van Daele, Locally compact quantum groups. A von Neumann algebra approach, preprint, 2006  Wells, B.B., Homomorphisms and translates of bounded functions, Duke math. J., 41, 35-39, (1974) · Zbl 0281.28004  Young, N.J., The irregularity of multiplication in group algebras, Q. J. math. Oxford ser., 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.