Towards the Carpenter's Theorem

TL;DR

This paper advances the understanding of Kadison's Carpenter's Theorem by demonstrating new cases where projections in a II_1 factor can be constructed with prescribed conditional expectations onto a masa.

Contribution

It establishes the existence of projections with prescribed conditional expectations under certain conditions, expanding the known instances of the Carpenter's Theorem.

Findings

Existence of projections with prescribed expectations for discrete elements in the masa

Counterexample for diffuse operators in the masa

Extension of the theorem to semiregular masas in separable factors

Abstract

Let M be a II_1 factor, A a masa in M and E the unique conditional expectation on A. Under some technical assumptions on the inclusion of A in M, which hold true for any semiregular masa of a separable factor, we show that for every discrete a in the positive part of the unit ball of A it is possible to find a projection p in M such that E(p)=a$. We also show an example of a diffuse operator x in A such that there exists a projection q in M with E(q)=x. These results show a new family of instances of a conjecture by Kadison, the so-called "Carpenter's Theorem".