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 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.

Multivariable Schur–Horn theorems

“…We then collect consequences of this theorem, including an approximate Schur-Horn theorem for general tuples of commuting hermitians and an approximate multivariable carpenter theorem. In section 3.4, we adapt an idea of Dykema et al [10] to show that we can find diffuse abelian orthogonal subalgebras of masas (in the sense of Popa) and use this to show the partial validity of Statement 6 in certain II 1 factors. In section 4.1, we digress to discuss the situation in matrix algebras, obtaining some approximate representations of the action of doubly stochastic matrices in terms of dilations.…”

“…In a recent paper [10], Dykema, Hadwin, Fang and Smith presented a different approach to the Schur-Horn and carpenter theorems, relating the problems to one involving the kernels of conditional expectations onto masas, when restricted to corners of type II 1 factors. This technique allows to improve a result of Popa [19] on orthogonal subalgebras of type II 1 factors.…”

Numerical Ranges in II1 Factors

“…Example 5.7. The quasinilpotent DT-operator S was introduced in [8] as one of an interesting class of operators in the free group factor L(F 2 ), that can be realized as limits of upper triangular random matrices. As the name suggests, its spectrum is {0}, and it satisfies S = √ e and τ (S * S) = 1/2.…”

A Measurable Selector in Kadison’s Carpenter’s Theorem

“…This research area has been initiated by Arveson and Kadison [7,31] who have asked for a characterization of D A (T ) when T is a projection, or more generally a self-adjoint operator, in a von Neumann factor of type II 1 . Problem 1 was investigated by a number of authors [2,3,4,10,24] and settled by Ravichandran [36,38]. The same problem when T is a normal…”