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Module homomorphisms and multipliers on locally compact quantum groups. (English) Zbl 1181.46036
The authors unify some existing results in abstract harmonic analysis related to module homomorphisms, multipliers and topological center in the context of locally compact quantum groups.
Let $$A$$ be a Banach algebra. For every $$\omega\in {A}$$ and $$x\in {A}^{\star}$$, define $$x\cdot\omega$$, $$\omega\cdot x \in {A}^{\star}$$ by $$x\cdot\omega(\nu)= x(\omega\nu)$$, $$\omega\cdot x(\nu)=x(\nu\omega)$$ $$(\nu\in A)$$. For every $$X \subseteq A^{\star}$$ and $$Y \subseteq A$$, set
$X\cdot Y=\{x\cdot\omega : x\in X,~\omega\in Y\},~Y\cdot X=\{\omega \cdot x : x\in X,~\omega\in Y\},$ and let $$\langle X\cdot Y\rangle$$, $$\langle Y\cdot X\rangle$$ denote the closed linear spans of $$X\cdot Y$$ and $$Y\cdot X$$, respectively.
If $$X$$ is a closed subspace of $$A^{\star}$$, for every $$m \in X^{\star}$$ and $$x \in X$$ define $$m \odot x \in A^{\star}$$ by $$m \odot x(\omega) = m (x \cdot \omega)$$ $$(\omega \in A)$$. Set
$\mathcal{Z}(X^{\star})=\{n\in X^{\star} : ~m\mapsto n\odot m \text{ is $$w^{\star}-w^{\star}$$-continuous on } X^{\star}\}.$ Let $$\mathbb{G}=(\mathfrak{M},\Gamma,\varphi,\psi)$$ be a von Neumann algebraic locally compact quantum group (for the basic definitions about quantum groups, see J. Kustermans and S. Vaes [Ann. Sci. Éc. Norm. Supér. (4) 33, No. 6, 837–934 (2000; Zbl 1034.46508)] and [Math. Scand. 92, No. 1, 68–92 (2003; Zbl 1034.46067)]).
Analogously to the locally compact group case, the authors set $$L^{\infty}(\mathbb{G})= \mathfrak{M}$$, $$L^{1}(\mathbb{G})= \mathfrak{M}_*$$, $$\text{LUC}(\mathbb{G}) = \langle L^{\infty}(\mathbb{G}) \cdot L^{1}(\mathbb{G}) \rangle$$, $$\text{RUC}(\mathbb{G}) = \langle L^{1}(\mathbb{G}) \cdot L^{\infty}(\mathbb{G}) \rangle$$ and write $$\text{WAP}(\mathbb{G})$$ for the weakly aperiodic elements of $$L^{1}(\mathbb{G})$$. Let $$(C_{0}(\mathbb{G}),\Gamma_c,\varphi_c,\psi_c)$$ denote the reduced $$C^{\star}$$-algebraic quantum group of $$\mathbb{G}$$, and set $$M(\mathbb{G})=C_{0}(\mathbb{G})^{\star}$$.
For every Banach algebra $$A$$ with an approximate identity bounded by one, the authors characterize the $$w^{\star}$$-$$w^{\star}$$-continuous $$A$$-module homomorphisms of $$A^{\star}$$. In particular, it is shown that that a co-amenable locally compact quantum group $$\mathbb G$$ is compact if and only if all $$L^{1}(\mathbb{G})$$-module homomorphisms of $$L^{\infty}(\mathbb{G})$$ are $$w^{\star}$$-$$w^{\star}$$-continuous; and that $$\mathbb G$$ is discrete if and only if all $$C_{0}(\mathbb G)$$-module homomorphisms of $$M(\mathbb G)$$ are $$w^{\star}$$-$$w^{\star}$$-continuous.
The authors show that for a co-amenable locally compact quantum group $$\mathbb G$$, the left and right multiplier algebras of $$L^{1}(\mathbb{G})$$ can be identified with $$M(\mathbb G)$$. It is obtained that there exists an isometric homomorphism $$\Theta$$ from $$M(\mathbb G)$$ into $$\text{LUC}(\mathbb{G})^\star$$ such that $$\Theta$$ is onto if and only if $$\mathbb G$$ is compact. This is used to show that $$\mathbb{G}$$ is compact if and only if $$\text{LUC}(\mathbb{G})=\text{WAP}(\mathbb{G})$$ and $$\Theta(M(\mathbb{G}))=\mathcal{Z}(\text{LUC}(\mathbb{G})^{\star})$$, which partially answers a question of V. Runde [J. Lond. Math. Soc., II. Ser. 80, No.  1, 55–71 (2009; Zbl 1188.46048)]. These results cover several related results of A. T.-M. Lau.

##### MSC:
 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 43A20 $$L^1$$-algebras on groups, semigroups, etc. 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 20G42 Quantum groups (quantized function algebras) and their representations
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