Schur–Horn theorems in II_∞-factors

Martín Argerami and Pedro Massey

Vol. 261 (2013), No. 2, 283–310

DOI: 10.2140/pjm.2013.261.283

Abstract

We describe majorization between selfadjoint operators in a σ-finite II_∞ factor (ℳ,τ) in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra 𝒜⊂ℳ that admits a (necessarily unique) trace-preserving conditional expectation, denoted by E_𝒜, we characterize the closure in the measure topology of the image through E_𝒜 of the unitary orbit of a selfadjoint operator in ℳ in terms of majorization (i.e., a Schur–Horn theorem). We also obtain similar results for the contractive orbit of positive operators in ℳ and for the unitary and contractive orbits of τ-integrable operators in ℳ.

Keywords

II_∞ factors, majorization, Schur–Horn theorem

Mathematical Subject Classification 2010

Primary: 46L51

Secondary: 46L10, 52A05, 15A18

Milestones

Received: 16 May 2011

Accepted: 9 October 2012

Published: 20 March 2013