Representations of logmodular algebras

TL;DR

This paper investigates the conditions under which contractive representations of logmodular algebras are completely contractive, extending known results and employing non-commutative operator space techniques.

Contribution

It proves that 2-contractive representations of logmodular algebras extend to positive maps on their C*-envelopes, generalizing previous results for uniform logmodular algebras.

Findings

2-contractive representations extend to positive maps

Generalization of Foias and Suciu's result

Matrix factorization results for uniform logmodular algebras

Abstract

We study the question of whether or not contractive representations of logmodular algebras are completely contractive. We prove that a 2-contractive representation of a logmodular algebra extends to a positive map on the enveloping C*-algebra, which we show generalizes a result of Foias and Suciu on uniform logmodular algebras. Our proof uses non-commutative operator space generalizations of classical results on 2-summing maps and semispectral measures. We establish some matrix factorization results for uniform logmodular algebras