Distance between unitary orbits in von Neumann algebras (Q1117433)

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 selfadjoint elements in Haagerup \(L^ p\)-spaces. The paper contains the appendix by H. Kosaki where the Powers-Størmer inequality is generalized to positive elements in Haagerup \(H^ p\)- spaces.