Multipliers of locally compact quantum groups via Hilbert \(C^*\)-modules.

*(English)*Zbl 1235.43004Author’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.

Reviewer: Ye Jiachen (Shanghai)

##### 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. |