Schur-Horn theorems in \(II_\infty\)-factors (Q1951193)

As the authors point out in the introduction, the classical Schur-Horn Theorem for matrices can be stated as follows: \[ \{x\in\mathbb{R}^n: x \prec y\}= \{ \text{diag}(UM_yU^*): U\in{\mathcal U}(n) \}, \] where {\parindent=6mm \begin{itemize}\item[(1)] \(x\prec y\) means that \(x\) is majorized by \(y\), i.e., if \(x\) and \(y\) are rearranged so as to have their coordinates in nonincreasing order, \(\sum_{j=1}^k x_j\leq \sum_{j=1}^k y_j\) for \(1\leq k\leq n-1\), and \(\sum_{j=1}^n x_j=\sum_{j=1}^n y_j\). \item[(2)] \(M_y\) is the diagonal matrix with \(y\) in the diagonal, \(\text{diag}(A)\) is the \(n\)-tuple of diagonal entries of \(A\), and \({\mathcal U}(n)\) is the unitary group of \(\mathbb{C}^n\). \end{itemize}} In this paper, the authors prove generalizations of the above formula in the context of a \(\sigma\)-finite II\(_\infty\) von Neumann factor \({\mathcal M}\) (with semifinite trace \(\tau\)). They define the following notion of majorization between selfadjoint elements \(a,b\) in \({\mathcal M}\): {\parindent=6mm \begin{itemize}\item[(a)] \(a\) is submajorised by \(b\), in symbols \(a\prec_w b\), if \(U_t(a)\leq U_t(b)\) for all \(t\geq 0\), where \[ U_t(x)=\int_0^t \lambda_s(x)\,ds, \] with \(\lambda_s(x)\) denoting the so-called upper spectral scale of the selfadjoint element \(x\in{\mathcal M}\), computed in terms of the spectral projections \(p^x\) of \(x\): \[ \lambda_s(x)=\min\{r\in\mathbb{R}: \tau(p^x(r,+\infty))\leq s\}, \;\;s\geq 0. \] \item[(b)] \(a\) is majorized by \(b\), in symbols \(a\prec_{w} b\), if \(a\prec b\) and \(L_t(a)\geq L_t(b)\) for all \(t\geq 0\), where \[ L_t(x)=\int_0^t \mu_s(x)\, ds, \] with \(\mu_s(x)\) denoting the lower spectral scale of \(x\), \(\mu_s(x)=-\lambda_s(-x)\). \end{itemize}} The main theorem of this paper (Theorem 5.5, Schur-Horn theorem for II\(_\infty\) factors) states the following: Let \({\mathcal A}\subset{\mathcal M}\) be a diffuse abelian von Neumann subalgebra that admits a unique trace preserving conditional expectation \(E_{\mathcal A}\). For any selfadjoint element \(b\in{\mathcal M}\), one has \[ \overline{E_{\mathcal A}({\mathcal U}_{\mathcal M}(b))}^{\mathcal T}=\{a\in{\mathcal A}: a^*=a, \;a\prec b\}. \] Here, \({\mathcal U}_{\mathcal M}(b)\) is the unitary orbit of \(b\) \(\{uau^*: u \text{ unitary in } {\mathcal M}\}\), and \({\mathcal T}\) is the measure topology of \({\mathcal M}\).