Eigenvalue inequalities in an embeddable factor

Bercovici, Hari; Li, Wing Suet

doi:10.1090/s0002-9939-05-07952-9

Cited by 10 publications

(8 citation statements)

References 9 publications

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“…The purpose of this paper is to prove that analogues of some of the Horn inequalities hold in all finite von Neumann algebras. This question was first considered by Bercovici and Li in [1] (see also [2]) and the following exposition is essentially from their papers. Let M be a von Neumann algebra with a fixed normal, faithful, tracial state τ .…”

Section: Introduction and Description Of Resultsmentioning

confidence: 99%

On a reduction procedure for Horn inequalities in finite von Neumann algebras

Collins¹,

Dykema²

2009

Operators and Matrices

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Abstract. We consider the analogues of the Horn inequalities in finite von Neumann algebras, which concern the possible spectral distributions of sums a + b of self-adjoint elements a and b in a finite von Neumann algebra. It is an open question whether all of these Horn inequalities must hold in all finite von Neumann algebras, and this is related to Connes' embedding problem. For each choice of integers 1 ≤ r ≤ n, there is a set T n r of Horn triples (I, J, K) of r-tuples of integers, and the Horn inequalities are in one-to-one correspondence with ∪ 1≤r≤n T n r . We consider a property P n , analogous to one introduced by Therianos and Thompson in the case of matrices, amounting to the existence of projections having certain properties relative to arbitrary flags, which guarantees that a given Horn inequality holds in all finite von Neumann algebras. It is an open question whether all Horn triples in T n r have property P n . Certain triples in T n r can be reduced to triples in T n−1 r by an operation we call TT-reduction. We show that property P n holds for the original triple if property P n−1 holds for the reduced one. A major part of this paper is devoted to showing that this operation of reduction preserves the value of the corresponding Littlewood-Richardson coefficients. We then characterize the TT-irreducible Horn triples in T n 3 , for arbitrary n, and for those LR-minimal ones (namely, those having Littlewood-Richardson coefficient equal to 1), we perform a construction of projections with respect to flags in arbitrary von Neumann algebras in order to prove property P n for them. This shows that all LR-minimal triples in ∪ n≥3 T n 3 have property P n , and so that the corresponding Horn inequalities hold in all finite von Neumann algebras.

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Section: Introduction and Description Of Resultsmentioning

confidence: 99%

On a reduction procedure for Horn inequalities in finite von Neumann algebras

Collins¹,

Dykema²

2009

Operators and Matrices

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“…It is, after a change of variable, the function defined by Murray and von Neumann [13,Lemma 15.2.1] and used in various forms by several authors (e.g. [12], [14], [4]). In terms of eigenvalue functions, the relation a b is characterized by the inequalities…”

Section: The Schur-horn Theoremmentioning

confidence: 99%

The carpenter and Schur–Horn problems for masas in finite factors

Dykema¹,

Fang²,

Hadwin³

et al. 2012

Illinois J. Math.

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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 ∈ A with 0 ≤ x ≤ 1 be expressed as E A (p) for a projection p ∈ M ?In this paper, we investigate these problems for various masas. We give positive solutions for the generator and radial masas in free group factors, and we also solve affirmatively a weaker form of the Schur-Horm problem for the Cartan masa in the hyperfinite factor.

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“…With this notation, the following result is a direct consequence of Theorem 2.1, and the reader is referred to [2] for a detailed proof: Corollary 2.2. Let a ∈ M be an Hermitian element, and {X (m) } m∈N a sequence of matricial approximations of a.…”

Section: Some Preliminaries On II 1 Factorsmentioning

confidence: 99%

“…Later, Horn's result was extended to the infinite dimensional setting by Bercovici et. al. in two papers [2,3] that deal with the case of operators in an embeddable II 1 factor and with compact operators respectively. Then, it is only natural to ask for extensions of Thompson's formula on adequate infinite dimensional settings.…”

Section: Introductionmentioning

confidence: 99%

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Thompson-type formulae

Antezana

Larotonda

Varela

2012

Journal of Functional Analysis

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Let X and Y be two n × n Hermitian matrices. In the article Proof of a conjectured exponential formula (Linear and Multilinear Algebra (19) 1986, 187-197) R. C. Thompson proved that there exist two n × n unitary matrices U and V such that e i X e i Y = e i U XU * +V BV * .In this note we consider extensions of this result to compact operators as well as to operators in an embeddable II 1 factor. 1

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Eigenvalue inequalities in an embeddable factor

Abstract: Abstract. We provide a characterization of the possible eigenvalues of the sum of two selfadjoint elements of a II 1 factor which can be embedded in the ultrapower R ω of the hyperfinite II 1 factor.

Cited by 10 publications

References 9 publications

On a reduction procedure for Horn inequalities in finite von Neumann algebras

On a reduction procedure for Horn inequalities in finite von Neumann algebras

The carpenter and Schur–Horn problems for masas in finite factors

Thompson-type formulae

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