Invariant means and multipliers on convolution quantum group algebras

Abstract

Let [Formula: see text] be a locally compact quantum group. Then the space [Formula: see text] of trace class operators on [Formula: see text] is a Banach algebra with the convolution induced by the right fundamental unitary of [Formula: see text]. We show that properties of [Formula: see text] such as amenability, triviality and compactness are equivalent to the existence of left or right invariant means on the convolution Banach algebra [Formula: see text]. We also investigate the relation between the existence of certain (weakly) compact right and left multipliers of [Formula: see text] and some properties of [Formula: see text].