Towards the carpenter’s theorem

Abstract

Abstract. Let M be a II 1 factor with trace τ , A ⊆ M a masa and E A the unique conditional expectation onto A. Under some technical assumptions on the inclusion A ⊆ M, which hold true for any semiregular masa of a separable factor, we show that for elements a in certain dense families of the positive part of the unit ball of A, it is possible to find a projection p ∈ M such that E A (p) = a. This shows a new family of instances of a conjecture by Kadison, the so-called "carpenter's theorem".

The Schur–Horn theorem for operators with three point spectrum

“…Other notable progress includes the work of Arveson [4] on diagonals of normal operators with finite spectrum. Moreover, Antezana, Massey, Ruiz, and Stojanoff [1] refined the results of Neumann [23], and Argerami and Massey [2,3] studied extensions to II 1 factors. For a detailed survey of recent progress on infinite Schur-Horn majorization theorems and their connections to operator ideals we refer to the paper of Kaftal and Weiss [18].…”

The Schur–Horn problem for normal operators

“…A MASA A ⊆ M is said to be totally complementable if, for every projection P ∈ A, the MASA AP of P MP admits a diffuse orthogonal abelian subalgebra. With the above definitions in hand, we are now able to establish a multivariate generalization of [2,Theorem 3.2]. We note that the proof is virtually identical.…”

Strong sums of projections in von Neumann factors

“…A positive operator A is said to be diagonalizable if A = γ j E j for some γ j > 0 and mutually orthogonal projections {E j } in M. Diagonalizable operators are also called discrete and are the most accessible operators in a type II factor (e.g., see [3]). …”