Thompson's theorem for II\(_1\) factors (Q2833026)

In [SIAM J. Appl. Math. 32, 39--63 (1977; Zbl 0361.15009)], \textit{R. C. Thompson} was able to characterize those pairs of vectors \(x\in\mathbb R^n\) and \(y\in\mathbb C^n\) with \(x_1\geq\cdots\geq x_n\geq0\) and \(|y_1|\geq\cdots\geq|y_n|\) such that there exists \(A\in M_n(\mathbb C )\) with singular values \(x\) and diagonal \(y\). The conditions are NEWLINE\[NEWLINE\begin{aligned} &\sum_{j=1}^k|y_j|\leq\sum_{j=1}^k x_j, \quad k=1,\dots,n, \\ &\sum_{j=1}^{n-1}|y_j|-|y_n|\leq\sum_{j=1}^{n-1}x_j-x_n. \end{aligned}\tag{1}NEWLINE\]NEWLINE This characterization is a variation of the Schur-Horn theorem, which characterizes those pairs in the case where \(A\) is selfadjoint. The inequalities (1) amount to what is known as submajorization (denoted by \(|y|\prec_w x\)).NEWLINENEWLINEThe existence of a matrix \(A\) with diagonal \(y\) and singular values \(x\) is equivalent to say that there exist unitaries \(U,V\) such that the diagonal of \(UYV\) is \(x\), where \(Y\) is the diagonal matrix with diagonal \(y\).NEWLINENEWLINEA natural framework where these ideas can be extended is II\(_1\)-factors. Much work has been done on this, and I refer to the paper's reference list for a subset of significant papers on the subject. The reason II\(_1\)-factors are amenable to these ideas lies in the fact that they are in a sense the ``right'' infinite-dimensional generalization of full matrix algebras (in particular, because they have a faithful trace, and continuous analogues of eigenvalues and singular values, which allow one to define majorization).NEWLINENEWLINESo here the authors prove a Thompson theorem in the setting of a II\(_1\)-factor. The accepted generalization of ``diagonal'' to a II\(_1\)-factor \(\mathcal M\) is a conditional expectation (i.e., norm-one projection) \(E_{\mathcal A}\) onto a masa \(\mathcal A\). The result proven by the authors is:NEWLINENEWLINETheorem. For a II\(_1\)-factor \(\mathcal M\) and masa \(\mathcal A\subset\mathcal M\) with conditional expectation \(E_{\mathcal A}\), and \(T\in\mathcal M\), \(A\in\mathcal A\), the following statements are equivalent: {\parindent=0.7cm\begin{itemize}\item[(i)] \(A\prec_w T\); \item[(ii)] \(A\in E_{\mathcal A}(\overline{\{UTV:\;U,V\in\mathcal U(\mathcal M)\}})\). NEWLINENEWLINE\end{itemize}} The proof, which comprises most of the paper after the introduction and preliminaries, consists of stricter reductions of the problem to simpler cases, and it uses in an essential way \textit{M.\,Ravichandran}'s Schur-Horn theorem in II\(_1\)-factors [``The Schur-Horn theorem in von Neumann algebras'', Preprint (2012), \url{arXiv:1209.0909}], which is basically the theorem above in the case where \(A\) and \(T\) are selfadjoint and \(V=U^*\).NEWLINENEWLINEIn a brief last section, the authors prove that their result implies Ravichandran's.