M. Argerami, P. Massey
Given a II$_1$ factor M and a masa A of M, we prove a version of the Schur-Horn Theorem, together with a contractive version. These results are inspired on a recent conjecture of Arveson and Kadison (math.OA/0508482).
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The carpenter problem in the context of $II_1$ factors, formulated by Kadison asks: Let $\mathcal{A} \subset \mathcal{M}$ be a masa in a type $II_1$ factor and let $E$ be the normal conditional expectation from $\mathcal{M}$ onto $\mathcal{A}$. Then, is it true that for every positive contraction $A$ in $\mathcal{A}$, there is a projection $P$ in $\mathcal{M}$ such that $E(P) = A$? In this note, we show that this is true if $A$ has finite spectrum. We will then use this result to prove an exact Schur-Horn theorem for (positive)operators with finite spectrum and an approximate Schur-Horn theore…
We consider the Schur-Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is well-known to be extremely difficult, and in fact, it remains open for matrices of size greater than $3$. We show that the infinite dimensional version of this problem is more tractable, and establish approximate solutions for normal operators in von Neumann factors of type I$_\infty$, II and III. A key result is an approximation theorem that can be seen as…
The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we intr…
The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences x and y that converge to 0, there exists a compact operator A with eigenvalue list y and diagonal sequence x if and only if y majorizes x (\sum_{j=1}^n x_j \le \sum_{j=1}^n y_j for all n) if and only if x = Qy for some orthostochastic matrix Q. The similar result requiring equality of the infinite series in the case that the sequences x and y are summable is an extension of a recent theorem by Arveson and Kadison. Our proof depends on the construction and …
Schur-Horn theorems focus on determining the diagonal sequences obtainable for an operator under all possible basis changes, formally described as the range of the canonical conditional expectation of its unitary orbit. Following a brief background survey, we prove an infinite dimensional Schur-Horn theorem for positive compact operators with infinite dimensional kernel, one of the two open cases posed recently by Kaftal- Weiss. There, they characterized the diagonals of operators in the unitary orbits for finite rank or zero kernel positive compact operators. Here we show how the characteriza…
A few years ago, Richard Kadison thoroughly analysed the diagonals of projection operators on Hilbert spaces and asked the following question: Let $\mathcal{A}$ be a masa in a type $II_1$ factor $\mathcal{M}$ and let $A \in \mathcal{A}$ be a positive contraction. Letting $E$ be the canonical normal conditional expectation from $\mathcal{M}$ to $\mathcal{A}$, can one find a projection $P \in \mathcal{M}$ so that [E(P) = A?] In a later paper, Kadison and Arveson, as an extension, conjectured a Schur-Horn theorem in type $II_1$ factors. In this paper, I give a proof of this conjecture of Arveson …
Two classical theorems in matrix theory, due to Schur and Horn, relate the eigenvalues of a self-adjoint matrix to the diagonal entries. These have recently been given a formulation in the setting of operator algebras as the Schur-Horn problem, where matrix algebras and diagonals are replaced respectively by finite factors and maximal abelian self-adjoint subalgebras (masas). There is a special case of the problem, called the carpenter problem, which can be stated as follows: for a masa A in a finite factor M with conditional expectation E_A, can each x in A with 0 <= x <= 1 be expressed as E_…
We describe majorization between selfadjoint operators in a $σ$-finite II$_\infty$ factor $(\mathcal{M},τ)$ in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra $\mathcal{A}\subset \mathcal{M}$ with trace-preserving conditional expectation $E_{\mathcal{A}}$, we characterize the closure in the measure topology of the image through $E_{\mathcal{A}}$ of the unitary orbit of a selfadjoint operator in $\mathcal{M}$ in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in $\mathcal{M}$ and …
Let $\mathcal A\subseteq \mat$ be a unital $*$-subalgebra of the algebra $\mat$ of all $n\times n$ complex matrices and let $B$ be an hermitian matrix. Let $\U_n(B)$ denote the unitary orbit of $B$ in $\mat$ and let $\mathcal E_\mathcal A$ denote the trace preserving conditional expectation onto $\mathcal A$. We give an spectral characterization of the set $$ \mathcal E_\mathcal A(\U_n(B))=\{\mathcal E_\mathcal A(U^* B U): U\in \mat,\ \text{unitary matrix}\}.$$ We obtain a similar result for the contractive orbit of a positive semi-definite matrix $B$. We then use these results to extend the n…
We prove a variety of results describing the possible diagonals of tuples of commuting hermitian operators in type $II_1$ factors. These results are generalisations of the classical Schur-Horn theorem to the infinite dimensional, multivariable setting. Our description of these possible diagonals uses a natural generalisation of the classical notion of majorization to the multivariable setting. In the special case when both the given tuple and the desired diagonal have finite joint spectrum, our results are complete. When the tuples do not have finite joint spectrum, we are able to prove strong…
A theorem of Thompson provides a non-self-adjoint variant of the classical Schur-Horn theorem by characterizing the possible diagonal values of a matrix with given singular values. We prove an analogue of Thompson's theorem for II_1 factors.
A theorem of Mirsky provides necessary and sufficient conditions for the existence of an N-square complex matrix with prescribed diagonal entries and prescribed eigenvalues. We give a simple inductive proof of this theorem.
C*-algebraic Schur product theorem, Pólya-Szegő-Rudin question and Novak's conjecture
本文将Schur乘积推广到任意C*-代数上，推导出Schur和Vybíral的结果，提出并解决了交换有单位C*-代数上的Novak猜想，同时提出了C*-代数Schur乘积的Pólya-Szegő-Rudin问题。
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