Wing Suet Li

Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor

The intersection ring of a complex Grassmann manifold is generated by Schubert varieties, and its structure is governed by the Littlewood–Richardson rule. Given three Schubert varieties S1, S2, S3 with intersection number equal to one, we show how to construct an explicit element in their intersection. This element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags, and is therefore well defined over an arbitrary field of scalars. Moreover, this result also applies to appropriately defined analogues of Schubert varieties in the Grassmann manifolds associated with a finite von Neumann algebra. The arguments require the combinatorial structure of honeycombs, particularly the structure of the rigid extremal honeycombs. It is known that the eigenvalue distributions of self-adjoint elements a,b,c with a+b+c=0 in the factor Rω are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. In particular, if x,y,z are self-adjoint elements of such a factor and x+y+z=0, then there exist self-adjoint a,b,c∈Rω such that a+b+c=0 and a (respectively, b,c) has the same eigenvalue distribution as x (respectively, y,z). A (‘complete’) matricial form of this result is known to imply an affirmative answer to an embedding question formulated by Connes. The critical point in the proof of this result is the production of elements in the intersection of three Schubert varieties. When the factor under consideration is the algebra of n×n complex matrices, our arguments provide new and elementary proofs of the Horn inequalities, which do not require knowledge of the structure of the cohomology of the Grassmann manifolds.

Eigenvalue inequalities in an embeddable factor

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 w of the hyperfinite II 1 factor.

Inequalities for eigenvalues of sums in a von Neumann algebra

We prove an analogue of the Freede—Thompson inequalities for the sum of two selfadjoint elements of a von Neumann algebra. The result also applies to singular values, and implies earlier results of Grothendieck and Fack.

Invariant subspaces of nilpotent operators and LR-sequences

The aim of this paper is to study systematically invariant subspaces of finitedimensional nilpotent operators. Our main motivation comes from classifying the similarity orbit in thelattice of invariant subspaces of a given nilpotent operator. We give a detailed study of the Littlewood-Richardson similarity orbit. We show that none of the “natural” similarity relations is equivalent with the others.

Characterization of singular numbers of products of operators in matrix algebras and finite von Neumann algebras

Abstract We characterize in terms of inequalities the possible generalized singular numbers of a product AB of operators A and B having given generalized singular numbers, in an arbitrary finite von Neumann algebra. We also solve the analogous problem in matrix algebras M n ( C ) , which seems to be new insofar as we do not require A and B to be invertible.

On Isometric Intertwining Liftings

The commutant lifting theorem of Sz.-Nagy and Foias ([19]), a cornerstone result in the theory of contractive operators on Hilbert space, has found a large variety of applications in most distinct areas of pure and applied analysis (see [13] and the references within) The existence theorem is complemented by different descriptions of the class of all commutant liftings of a given triple. Subsequently, a major direction of studies have concentrated on describing the properties of distinguished liftings; as interesting cases, let us note the central lifting([14], [8]), or the liftings which correspond in engineering applications to rational realizations([18]).

Classical linear algebra inequalities for the Jordan models of C0 operators

Abstract We extend to the class C 0 certain inequalities, relating the sizes of the Jordan blocks of a matrix with those of a restriction/compression of the matrix. The proofs seem to be more natural than those in the original linear algebra approach.