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arXiv · 1110.6500

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

Ken Dykema·Junsheng Fang·Don Hadwin·Roger Smith
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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.

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Ken Dykema, Junsheng Fang, Don Hadwin, Roger Smith. 2011-10-29. The carpenter and Schur--Horn problems for masas in finite factors. https://arxiv.org/abs/1110.6500

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