Distance between unitary orbits in von Neumann algebras

Fumio Hiai and Yoshihiro Nakamura

Vol. 138 (1989), No. 2, 259–294

DOI: 10.2140/pjm.1989.138.259

Abstract

Let ℳ be a semifinite factor. For normal operators x and y in ℳ, introducing the spectral distance δ(x,y), we show that δ(x,y) ≥ dist(𝒰(x),𝒰(y)) ≥ c⁻¹δ(x,y) with a universal constant c, where dist(𝒰(x),𝒰(y)) denotes the distance between the unitary orbits 𝒰(x) and 𝒰(y). The equality dist(𝒰(x),𝒰(y)) = δ(x,y) holds in several cases. Submajorizations are established concerning the spectral scales of τ-measurable selfadjoint operators affiliated with ℳ. Using these submajorizations, we obtain the formulas of L^p-distance and anti-L^p-distance between unitary orbits of τ-measurable selfadjoint operators in terms of their spectral scales. Furthermore the formulas of those distances in Haagerup L^p-spaces are obtained when ℳ is a type III₁ factor. The appendix by H. Kosaki is the generalized Powers-Størmer inequality in Haagerup L^p-spaces.

Mathematical Subject Classification 2000

Primary: 46L10

Secondary: 46L50

Milestones

Received: 15 July 1987

Revised: 6 July 1988

Published: 1 June 1989