Complete boundedness of multiple operator integrals

TL;DR

This paper characterizes when multiple operator integrals are bounded on the Haagerup tensor product, showing they are automatically completely bounded and linking this to a factorization property of the symbol, generalizing previous results.

Contribution

It provides a new characterization of bounded multiple operator integrals, extending known results to a broader class of mappings and establishing their complete boundedness.

Findings

Bounded multiple operator integrals are automatically completely bounded.

A factorization property of the symbol characterizes boundedness.

Generalization of the Juschenko-Todorov-Turowska result on multilinear Schur multipliers.

Abstract

In this paper, we characterize the multiple operator integrals mappings which are bounded on the Haagerup tensor product of spaces of compact operators. We show that such maps are automatically completely bounded and prove that this is equivalent to a certain factorization property of the symbol associated to the operator integral mapping. This generalizes a result by Juschenko-Todorov-Turowska on the boundedness of continuous multilinear Schur multipliers.