Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

TL;DR

This paper develops a noncommutative theory of representing measures for characters of function algebras using von Neumann algebra conditional expectations, extending classical measure concepts into the noncommutative setting.

Contribution

It introduces a noncommutative Hoffman-Rossi theorem and noncommutative Jensen measures, advancing the understanding of representing measures in noncommutative analysis.

Findings

Existence criteria for noncommutative representing measures

Extension of classical measure theorems to noncommutative algebras

Identification of Jensen measures in the noncommutative context

Abstract

We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a `noncommutative Hoffman-Rossi theorem' giving the existence of weak* continuous `noncommutative representing measures' for so-called D-characters. These results may also be viewed as `module' Hahn-Banach extension theorems for weak* continuous `characters' into possibly noninjective von Neumann algebras. In closing we introduce the notion of `noncommutative Jensen measures', and show that as in the classical case representing measures of logmodular algebras are Jensen measures. The proofs of the two main cycles of results rely on the delicate interplay of Tomita-Takesaki theory, noncommutative Radon-Nikodym derivatives, Connes cocycles, Haagerup noncommutative Lp-spaces, Haagerup's reduction theorem, etc.