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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
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