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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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