Lyudmila Turowska: Herz-Schur multipliers of dynamical systems

We extend the notion of Herz-Schur multipliers to the setting of non-commutative dynamical systems: given a \(C^*\)-algebra \(A\), a locally compact group \(G\), and an action \(\alpha\) of\( G\) on \(A\), we define transformations on the (reduced) crossed product \(A\rtimes_{r,\alpha} G\) of \(A\) by \(G\), which, in the case \(A = \mathbb{C}\), reduce to the classical Herz-Schur multipliers. We shall also introduce Schur \(A\)-multipliers which, in the case \(A = \mathbb{C}\), reduce to the classical measurable Schur multipliers. We establish characterisations of Schur \(A\)-multipliers which generalise the classical descriptions of Schur multipliers and  present a transference theorem in the new setting, identifying isometrically the Herz-Schur multipliers of the dynamical system \((A,G,\alpha)\) with the invariant part of the Schur \(A\)-multipliers.  We shall discuss special classes of Herz-Schur multipliers, in particular, those which are associated to a locally compact abelian group \(G\) and its canonical action on the \(\) \(C^*\)-algebra \(C^*(\Gamma)\) of the dual group \(\Gamma\).
 
This is  a joint work with Andrew McKee and Ivan Todorov.