Central and convolution Herz-Schur multipliers

TL;DR

This paper characterizes central operator-valued Schur and Herz-Schur multipliers, linking them to bilinear multipliers and quantum group multipliers, with applications to contractive and idempotent cases.

Contribution

It provides new descriptions of central multipliers, connects them to quantum group multipliers, and characterizes contractive and idempotent multipliers in these classes.

Findings

Descriptions of central operator-valued Schur and Herz-Schur multipliers.

Identification of convolution multipliers as right completely bounded multipliers.

Characterization of contractive and idempotent multipliers.

Abstract

We obtain descriptions of central operator-valued Schur and Herz-Schur multipliers, akin to a classical characterisation due to Grothendieck, that reveals a close link between central (linear) multipliers and bilinear multipliers into the trace class. Restricting to dynamical systems where a locally compact group acts on itself by translation, we identify their convolution multipliers as the right completely bounded multipliers, in the sense of Junge-Neufang-Ruan, of a canonical quantum group associated with the underlying group. We provide characterisations of contractive idempotent operator-valued Schur and Herz-Schur multipliers. Exploiting the link between Herz-Schur multipliers and multipliers on transformation groupoids, we provide a combinatorial characterisation of groupoid multipliers that are contractive and idempotent.