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Multipliers of locally compact quantum groups via Hilbert $$C^*$$-modules. (English) Zbl 1235.43004
Author’s abstract: A result of Gilbert shows that every completely bounded multiplier $$f$$ of the Fourier algebra $$A(G)$$ arises from a pair of bounded continuous maps $$\alpha , \beta : G \rightarrow K$$, where $$K$$ is a Hilbert space, and $$f(s^{-1}t) = (\beta (t) | \alpha (s))$$ for all $$s, t \in G$$. We recast this in terms of adjointable operators acting between certain Hilbert $$C^*$$-modules, and show that an analogous construction works for completely bounded left multipliers of a locally compact quantum group. We find various ways to deal with right multipliers: one of these involves looking at the opposite quantum group, and this leads to a proof that the (unbounded) antipode acts on the space of completely bounded multipliers in a way that interacts naturally with our representation result. The dual of the universal quantum group (in the sense of Kustermans) can be identified with a subalgebra of the completely bounded multipliers, and we show how this fits into our framework. Finally, this motivates a certain way of dealing with two-sided multipliers.

##### MSC:
 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 46L08 $$C^*$$-modules 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 22D15 Group algebras of locally compact groups 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations 22D35 Duality theorems for locally compact groups 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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