A Schur-Horn theorem in II$_1$ factors

Abstract

Abstract. Given a II 1 factor M and a diffuse abelian von Neumann subalgebra A ⊂ M, we prove a version of the Schur-Horn theorem, namelywhere ≺ denotes spectral majorization, E A the unique trace-preserving conditional expectation onto A, and U M (b) the unitary orbit of b in M. This result is inspired by a recent problem posed by Arveson and Kadison.

A contractive version of a Schur–Horn theorem in II1 factors

“…In [5], Arveson and Kadison posed a (strong) version of the Schur-Horn theorem for II 1 factors as a problem and proved related results. As a first step toward settling the Arveson-Kadison problem, the authors have proven in [3] a weaker version, related with the point of view developed in [13]. In this note we obtain a weak contractive version of submajorization within II 1 factors in the spirit of [3] (Theorem 3.4).…”

“…Some historical aspects of the theory of majorization are mentioned in [3,5]. The Schur-Horn theorem, coined in the papers [8,15], is probably the most remarkable among the many characterizations known for these notions (see the precise statement of the theorem after Proposition 2.3).…”

Strong sums of projections in von Neumann factors

“…As in [2], the most tractable operators in a type II case are the diagonalizable ones. Applying Lemma 2.1 to a diagonalizable operator, the strong convergence of the "remainders" is obtained by showing that they converge in the trace-norm (Lemma 5.1).…”

Towards the carpenter’s theorem

“…In [1] we proved a weaker version of Arveson-Kadison's conjecture, which restricted to the situation in (1) is…”

“…Whether the Schur-Horn theorem extends or not to II 1 factors is unknown at the moment (see [1,3]). In this paper we focus on the CT in II 1 factors.…”