Majorization and Stochastic maps in von Neumann algebras
Let \({\mathcal M}\) be a semifinite von Neumann algebra with faithful normal semifinite trace \(\tau\) and \(\tilde {\mathcal M}\) the set of \(\tau\)- measurable operators affiliated with \({\mathcal M}\). Let \({\mathcal S}\) be the set of \(\tau\)-compact operators in \({\mathcal M}\) and \(\tilde {\mathcal S}\) the closure of \({\mathcal D}\) in \(\tilde {\mathcal M}\) in the measure topology. Between \(x,y\in \tilde {\mathcal M}_+\), the spectral relations of majorization \(y<x\), submajorization \(y\underset {.} < x\), spectral dominance \(y\lesssim x\) and spectral equivalence \(y\approx x\) are introduced by means of the generalized s-numbers of x,y. When \({\mathcal M}\) is a factor, the following are proved: (1) for \(x,y\in L^ 1({\mathcal M})_+\), \(y<x\) if and only if y is in the \(\| \cdot \|_ 1\)-closure of the convex hull of \(U(x)=\{uxu^*:u\in {\mathcal M}\) unitary\(\}\) ; (2) for \(x,y\in (L^ 1({\mathcal M})+{\mathcal S})_+\), \(y\underset {.} < x\) if and if y is in the closure of the convex hull of \(C(x)=\{axa^*:a\in {\mathcal M},\| a\| \leq 1\}\) in the measure topology; (3) for \(x,y\in (L^ 1({\mathcal M})+{\mathcal S})_+\), \(y\lesssim x\) if and only if y is in the closure of \(C(x)\) in the measure topology; (4) for \(x,y\in \tilde {\mathcal S}_+\), \(y\approx x\) if and only if y is in the closure of U(x) in the measure topology. The relation between the (sub)majorization and doubly (sub)stochastic maps is established. For instance, for \(x,y\in L^ 1({\mathcal M})_+\) with \(\tau (1)<\infty\), \(y<x\) if and only if \(y=\phi(x)\) for some doubly stochastic maps \(\phi\) on \({\mathcal M}\), and for \(x,y\in \tilde {\mathcal S}_+\), \(y\underset {.} < x\) if and only if \(y=\phi(x)\) for some double substochastic map \(\phi\) on \({\mathcal M}\). Finally the doubly substochastic maps \(\phi\) on \({\mathcal M}\) such that \(\phi(x)\approx x\) for all \(x\in {\mathcal M}\) are characterized in several ways.
scientific article; zbMATH DE number 4032395
scientific article