In this paper we obtain descriptions of central operator-valued Schur and Herz-Schur multipliers, akin to a classical characterisation due to Grothéndieck, 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.

We would like to thank Przemyslaw Ohrysko for helpful conversations during the preparation of this paper. The second author was partially supported by the Ministry of Sciences of Iran and School of Mathematics of Institute for research in fundamental sciences (IPM). Most of this work was completed when the second author was visiting Chalmers University of Technology. She would like to thank the Department of Mathematical Sciences of Chalmers University of Technology and the University of Gothenburg for warm hospitality. She would also like to thank Alireza Medghalchi and Massoud Amini for their support and encouragement during this work. The third author was supported by Simons Foundation Grant 708084.