## Illinois Journal of Mathematics

### The carpenter and Schur–Horn problems for masas in finite factors

#### 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 $\mathbb{E}_{A}$, can each $x\in A$ with $0\leq x\leq1$ be expressed as $\mathbb{E}_{A}(p)$ 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.

#### Article information

Source
Illinois J. Math., Volume 56, Number 4 (2012), 1313-1329.

Dates
First available in Project Euclid: 6 May 2014

https://projecteuclid.org/euclid.ijm/1399395834

Digital Object Identifier
doi:10.1215/ijm/1399395834

Mathematical Reviews number (MathSciNet)
MR3231485

Zentralblatt MATH identifier
1292.46040

Subjects
Primary: 46L10: General theory of von Neumann algebras
Secondary: 15A42: Inequalities involving eigenvalues and eigenvectors

#### Citation

Dykema, Kenneth J.; Fang, Junsheng; Hadwin, Donald W.; Smith, Roger R. The carpenter and Schur–Horn problems for masas in finite factors. Illinois J. Math. 56 (2012), no. 4, 1313--1329. doi:10.1215/ijm/1399395834. https://projecteuclid.org/euclid.ijm/1399395834

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