Multipliers of locally compact quantum groups via Hilbert C${}^{\ast }$-modules

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 $\alpha ,\beta :G\to K$, 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${}^{\ast }$-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 which 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 to deal with two-sided multipliers.

24 pages; many typos corrected; some rewriting

Cited by in corpus (6)

• The Haagerup property for locally compact quantum groups
• CCAP for universal discrete quantum groups
• Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms
• Completely positive multipliers of quantum groups
• Categorical aspects of quantum groups: multipliers and intrinsic groups
• The approximation property for locally compact quantum groups