Distance between unitary orbits in von Neumann algebras
type: Item
instance of: scholarly article
title: Distance between unitary orbits in von Neumann algebras
zbMATH Open document ID: 0667.46044
author: Fumio Hiai
author: Yoshihiro Nakamura
published in: Pacific Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pjm/1102650151
review text: Let \({\mathcal M}\) be a semifinite von Neumann algebra with a fixed faithful normal semifinite trace \(\tau\). Let \({\mathcal U}\) be the set of all unitaries in \({\mathcal M}\) and \({\mathcal U}(x)\) the unitary orbit \(\{uxu^*:\) \(u\in {\mathcal U}\}\) of \(x\in {\mathcal M}\). For normal elements \(x,y\in {\mathcal M}\), the spectral distance \(\delta\) (x,y) is introduced by comparing the traces of spectral projections of x and y. When \({\mathcal M}\) is a \(\sigma\)- finite semifinite factor, we show that \(\delta\) (x,y)\(\geq dist({\mathcal U}(x),{\mathcal U}(y))\geq c^{-1}\delta (x,y)\) for all normal elements \(x,y\in {\mathcal M}\) with a universal constant \(c>0\). The equality dist (\({\mathcal U}(x),{\mathcal U}(y))=\delta (x,y)\) is established for several cases. Let \(\tilde {\mathcal M}_{sa}\) be the set of all \(\tau\)-measurable operators affiliated with \({\mathcal M}\). When \(\tau (1)<\infty\), for \(x\in \tilde {\mathcal M}_{sa}\) with the spectral decomposition \(x=\int^{\infty}_{-\infty}sde_ s\), the spectral scale \(\lambda\) (x) of x is defined by \(\lambda_ t(x)=\inf \{s\in {\mathbb{R}}:\) \(\tau (1-e_ s)\leq t\}\) for \(t\in (0,\tau (1))\). By using the majorization method, we obtain the following formulas of \(L^ p\)-distance and anti-\(L^ p\)- distance between unitary orbits: when \({\mathcal M}\) is a finite factor, for \(x,y\in \tilde {\mathcal M}_{sa}\) and \(1\leq p\leq \infty\) \[ \inf_{u\in {\mathcal U}}\| x-uyu^*\|_ p=\| \lambda (x)-\lambda (y)\|_ p,\quad \sup_{u\in {\mathcal U}}\| x-uyu^*\|_ p=\| \lambda (x)+\lambda (-y)\|_ p. \] When \({\mathcal M}\) is an infinite semifinite factor, the analogous formulas of \(L^ p\)-distances between unitary orbits are obtained for x, y in a certain subclass of \(\tilde {\mathcal M}_{sa}\) with the modified spectral scales. Finally, when \({\mathcal M}\) is a factor of type \(III_ 1\), we exactly estimate the \(L^ p\)-distance and the anti-\(L^ p\)-distance between unitary orbits of se…
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
zbMATH DE Number: 4092146
zbMATH Keywords: semifinite von Neumann algebra with a fixed faithful normal semifinite trace
zbMATH Keywords: unitary orbit
zbMATH Keywords: spectral distance
zbMATH Keywords: traces of spectral projections
zbMATH Keywords: \(\sigma\)-finite semifinite factor
zbMATH Keywords: measurable operators
zbMATH Keywords: spectral decomposition
zbMATH Keywords: anti-\(L^ p\)-distance
zbMATH Keywords: selfadjoint elements in Haagerup \(L^ p\)-spaces
zbMATH Keywords: Powers-Størmer inequality
MaRDI profile type: Publication
OpenAlex ID: W2009897553
DOI: 10.2140/PJM.1989.138.259
description: scientific article; zbMATH DE number 4092146