The carpenter and Schur--Horn problems for masas in finite factors

TL;DR

This paper extends classical matrix theorems to operator algebras, solving the Schur-Horn and carpenter problems for specific masas in finite factors, advancing understanding in operator algebra theory.

Contribution

It provides new solutions to the Schur-Horn and carpenter problems for certain masas in finite factors, including free group factors and the hyperfinite factor.

Findings

Positive solutions for generator and radial masas in free group factors

Affirmative solution for a weaker Schur-Horn problem in the hyperfinite factor

Advancement in understanding of masas in finite operator algebras

Abstract

Two classical theorems in matrix theory, due to Schur and Horn, relate the eigenvalues of a self-adjoint matrix to the diagonal entries. These have recently been given a formulation in the setting of operator algebras as the Schur-Horn problem, where matrix algebras and diagonals are replaced respectively by finite factors and maximal abelian self-adjoint subalgebras (masas). There is a special case of the problem, called the carpenter problem, which can be stated as follows: for a masa A in a finite factor M with conditional expectation E_A, can each x in A with 0 <= x <= 1 be expressed as E_A(p) for a projection p in M? In this paper, we investigate these problems for various masas. We give positive solutions for the generator and radial masas in free group factors, and we also solve affirmatively a weaker form of the Schur-Horm problem for the Cartan masa in the hyperfinite factor.