Compactness properties of operator multipliers

TL;DR

This paper advances the understanding of operator multipliers by introducing the symbol concept, characterising compactness, and exploring their properties in tensor products of C*-algebras.

Contribution

It introduces the symbol of an operator multiplier, provides a complete characterisation of compact multipliers, and generalises results on multilinear maps in C*-algebras.

Findings

Characterisation of completely compact operator multipliers via their symbols

Conditions under which compact and completely compact multipliers coincide

Description of multilinear maps on C*-algebras of compact operators

Abstract

We continue the study of multidimensional operator multipliers initiated in [arXiv:math/0701645]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C*-algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C*-algebra of compact operators in terms of tensor products, generalising results of Saar.