THE CARPENTER AND SCHUR-HORN PROBLEMS FOR MASAS IN FINITE FACTORS

摘要

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 ∈ A with 0 ≤ x ≤ 1 be expressed as E_A (p) for a projection p ∈ 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.