Multivariable Schur-Horn theorems (Q2795907)

The classical Schur-Horn theorem states that for vectors \(\{d_i\}_{i=1}^N\) and \(\{\lambda_i\}_{i=1}^N\) in \(\mathbb{R}^N\) whose entries are in non-increasing order, there is a Hermitian matrix with diagonal \(\{d_i\}_{i=1}^N\) and eigenvalues \(\{\lambda_i\}_{i=1}^N\) if and only if NEWLINE\[NEWLINE\sum_{i=1}^nd_i\leq \sum_{i=1}^n\lambda_i, \quad n=1,2,\dots,N, NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sum_{i=1}^Nd_i= \sum_{i=1}^N\lambda_i.NEWLINE\]NEWLINE The above majorization inequalities can be extended for self-adjoint tuples:NEWLINENEWLINEDefinition. Given self-adjoint tuples \(\mathbf{S}=(S_1,\dots,S_n)\) and \(\mathbf{A}=(A_1,\dots,A_n)\) in a type II\(_1\) factor or a matrix algebra \(M\), we say that \(\mathbf{A}\) is majorized by \(\mathbf{S}\) (denoted by \(\mathbf{A}\prec\mathbf{S}\)) if there is a doubly stochastic map \(D\) on \(\mathcal{M}\) (that is, a linear, unital, positive, trace preserving map on \(\mathcal{M}\)) such that \(D(S_i)=A_i\) for \(1\leq i\leq n\).NEWLINENEWLINEThe main result in the paper under review is that, although the multivariable Schur-Horn theorem fails in general, it does hold in type II\(_1\) factors, under the additional assumption that the tuples involved both have finite spectrum. More precisely:NEWLINENEWLINETheorem. Let \(\mathcal{M}\) be a type II\(_1\) factor, \(\mathcal{A}\) be a masa in \(\mathcal{M}\) and let \(E\) be the trace preserving conditional expectation from \(\mathcal{M}\) onto \(\mathcal{A}\). Let \(\mathbf{A}\) and \(\mathbf{S}\) be \(n\)-tuples of commuting Hermitian operators with \(\mathbf{A}\in \mathcal{A}^n\) and \(\mathbf{S}\in \mathcal{M}^n\) such that \(\mathbf{A}\prec \mathbf{S}\). Moreover, assume that both \(\mathbf{A}\) and \(\mathbf{S}\) have finite spectrum. Then, there is a tuple \(\mathbf{T}\) in the norm closure of the unitary orbit of \(\mathbf{S}\), such that NEWLINE\[NEWLINEE(\mathbf{T})=\mathbf{A}.NEWLINE\]NEWLINE As a consequence, under the finite-spectrum assumption, the multivariable carpenter theorem also holds in type II\(_1\) factors:NEWLINENEWLINETheorem. Let \(\mathcal{M}\) be a type II\(_1\) factor, \(\mathcal{A}\) a masa in \(\mathcal{M}\) and let \(E\) be the trace preserving conditional expectation from \(\mathcal{M}\) onto \(\mathcal{A}\). Let \(\mathbf{A}\) be an \(n\)-tuples of positive contractions in \(\mathcal{A}\) with finite spectrum. Then there is an \(n\)-tuple of commuting projections \(\mathbf{P}\) in \(\mathcal{M}\) such that NEWLINE\[NEWLINEE(\mathbf{P})=\mathbf{A}.NEWLINE\]NEWLINE These results partially generalize the second author's results in the single variable case.