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

Abstract

We prove a contractive version of the Schur-Horn theorem for submajorization in II 1 factors that complements some previous results on the Schur-Horn theorem within this context. We obtain a reformulation of a conjecture of Arveson and Kadison regarding a strong version of the Schur-Horn theorem in II 1 factors in terms of submajorization and contractive orbits of positive operators.

Schur–Horn theorems in II∞-factors

“…generalizations of (1.2)) in the case of a σ-finite II ∞ -factor. These results extend those obtained in [2,3,26]. Our results are in the vein of Neumann's work, and they are related with a weak version of Arveson-Kadison's scheme for Schur-Horn theorems, but modeled in II ∞ factors.…”

“…Kadison conjectured a Schur-Horn theorem in II 1 factors. Although this conjecture remains an open problem, there has been progress on related (but weaker) Schur-Horn theorems in this context [2,3,5]. There has also been significant improvements of Neumann's work on majorization between sequences in c 0 (R + ) due to V. Kaftal and G. Weiss [21,22] because of the relations between infinite dimensional versions of the Schur-Horn theorem (via majorization of bounded structured real sequences) and arithmetic mean ideals (see also [7] for improvements in the compact case in B(H)).…”

“…Our results are in the vein of Neumann's work, and they are related with a weak version of Arveson-Kadison's scheme for Schur-Horn theorems, but modeled in II ∞ factors. These extensions are formally analogous to the Schur-Horn theorems in [2,3], but the techniques are more involved in the infinite case. We show that our results are optimal, in the sense that they can not be strengthened for a general selfadjoint operator in a II ∞ factor.…”

Submajorization on $\ell^p(I)^+$ determined by increasable doubly substochastic operators and its linear preservers

“…In recent years, there is a big progress towards developing various extensions of the most important majorization relations on sequence spaces [5,15] and on descrete Lebesgue spaces [6,7,8,9,10,18,19,20,21,22,23] with apropriate generalizations of some famous theorems in linear algebra [2,3,4,16,24,27,30,32]. There are a lot of applications of the majorization theory in various branches of mathematics and there exist significant conections with the other science like physics, quantum mechanics and quantum information theory [12,17,25,30,31,33].…”

The Paulsen Problem in operator theory

“…This is equivalent in frame theory to characterizing the sequences which occur as the norms of a frame with a specified frame operator. We refer the reader to [2,3,4,5,8,13,14,10,19,22,25,26,27,28] for a review of the work in this direction.…”