Measurable vectors for von Neumann algebras

Let A be a semifinite von Neumann algebra, with countably decomposable center, on the Hilbert space H. A measurable vector is a linear functional on H whose domain contains a strongly dense domain and which satisfies certain continuity conditions. H can be embedded as a dense subspace of the topological vector space of measurable vectors. The measurable vectors are a module over the measurable operators, and the action of measurable operators on measurable vectors is jointly continuous with respect to suitable topologies. If A is standard, then the measurable operators and measurable vectors are isomorphic as topological vector spaces. If the center of A is not countably decomposable, the results hold with minor changes.     

Transformations on tensor spaces II

We prove that a linear map of one tensor product space to another sending decomposable tensors to decomposable tensors is essentially a tensor product of linear maps of products of component factors of the domain into a selection of the factors of the range. The product of those factors of the domain not involved in the above is collapsed via a linear functional, and those factors in the range left out of the above provide a common factor in the range. In the statement of the main theorem the flanking maps are induced by permutations of the factors of the domain and the range and they present the products in manageable form. It is assumed that the underlying field has at least five members, but the necessity of this assumption is not settled. All vector spaces are finite dimensional and the tensor products have finitely many components.     

Chapter 2:rank and tensor rank

Let Vbe a vector space of matrices over a field and ka fixed positive integer. In this chapter we first survey results concerning linear maps on certain types of Vthat preserve one of the following:(a) the set of rank kmatrices, (b) the set of matrices of rank less than k. We next survey results concerning linear maps on certain symmetry classes of tensors that preserve nonzero decomposable elements.     

Lower bounds for the norms of decomposable symmetrized tensors

Lower bounds are given for the difference of two decomposable symmetrized tensors. The first bound uses a norm which makes the component vectors in a decomposable symmetrized tensor part of an orthonormal basis. The second bound holds only for decomposable elements of symmetry classes whose associated characters are linear.     

A class of generalized positive linear maps on matrix algebras

We construct a class of positive linear maps on matrix algebras. We find conditions when these maps are atomic, decomposable and completely positive. We obtain a large class of atomic positive linear maps. As applications in quantum information theory, we discuss the structural physical approximation and optimality of entanglement witness associated with these maps.     

A note on linear preservers of certain ranks of tensor products of matrices

In this note, we give a simple proof as well as an extension of a very recent result of B. Zheng, J. Xu and A. Fosner concerning linear maps between vector spaces of complex square matrices that preserve the rank of tensor products of matrices by using a structure theorem of R. Westwick on linear maps between tensor product spaces that preserve non-zero decomposable elements.     

On Bounded Local Resolvents

It is known that each normal operator on a Hilbert space with nonempty interior of the spectrum admits vectors with bounded local resolvent. We generalize this result for Banach space operators with the decomposition property (δ) (in particular for decomposable operators). Moreover, the same result holds for operators with interior points in the localizable spectrum.     

Irreducible positive linear maps on operator algebras

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible positive linear maps on von Neumann algebras are explicitly constructed, and a criterion for the irreducibility of decomposable positive maps on full matrix algebras is given.

     

On freely decomposable maps

Freely decomposable and strongly freely decomposable maps were introduced by G.R. Gordh and C.B. Hughes as a generalization of monotone maps with the property that these maps preserve local connectedness in inverse limits. We study further these types of maps, generalize some of the results by Gordh and Hughes and present examples showing that no further generalization is possible.     

Indecomposable positive linear maps constructed from permutations

We generalize two classes of D-type positive linear maps by permutations. For the first class, we discuss their indecomposability and give conditions when they are indecomposable and decomposable. For the other class, we show that they are atomic.     

张量空间中锥的可分解性

在文献 [1 ]的基础上研究张量空间中锥的性质 ,得到了张量空间中射影锥的极端向量的表示形式 .给出了张量空间中射影锥可分解的充分必要条件 ,并由此可得出有限维实空间中真正锥可分解的已有结论 .     

Decomposable symmetric mappings between infinite-dimensional spaces

Decomposable mappings from the space of symmetric k-fold tensors over E, , to the space of k-fold tensors over F, , are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable mapping belongs to a given operator ideal, then so does its inducing operator. This result allows us to classify injective linear operators between spaces of homogeneous approximable polynomials and between spaces of nuclear polynomials which map rank-1 polynomials to rank-1 polynomials.     

Decomposable subspaces,linear sections of Grassmann varieties,and higher weights of Grassmann codes

We consider the question of determining the maximum number of points on sections of Grassmannians over finite fields by linear subvarieties of the Plücker projective space of a fixed codimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. A basic tool used is a characterization of decomposable subspaces of exterior powers, that is, subspaces in which every nonzero element is decomposable. Also, we use a generalization of the Griesmer–Wei bound that is proved here for arbitrary linear codes.     

Facial Structures for Decomposable Positive Linear Maps in Matrix Algebras

We study the facial structures of the cone of all decomposable positive linear maps from the matrix algebra M_m into M_n. Especially, we completely determine the faces of the cone which arise from the dual of positive linear maps.Partially supported by BSRI-MOE and RIM-SNU     

Roots of almost decomposable operators

The author shows that, for an injective analytic function f, f(T) is almost decomposable iff T is almost decomposable, where T is a bounded linear operator on a Banach space and f(T) is defined by the functional calculus.     

Symmetric Hermitian decomposability criterion,decomposition, and its applications

The Hermitian tensor is an extension of Hermitian matrices and plays an important role in quantum information research. It is known that every symmetric tensor has a symmetric CP-decomposition. However, symmetric Hermitian tensor is not the case. In this paper, we obtain a necessary and sufficient condition for symmetric Hermitian decomposability of symmetric Hermitian tensors. When a symmetric Hermitian decomposable tensor space is regarded as a linear space over the real number field, we also obtain its dimension formula and basis. Moreover, if the tensor is symmetric Hermitian decomposable, then the symmetric Hermitian decomposition can be obtained by using the symmetric Hermitian basis. In the application of quantum information, the symmetric Hermitian decomposability condition can be used to determine the symmetry separability of symmetric quantum mixed states.     

Linear mappings on second symmetric product spaces that Preserve rank less than or equal to one

In this paper we characterize those linear mappings from a second symmetric product space to another which preserve decomposable elements of the form λu⋅u where u is a vector and λ is a scalar. This leads to the corresponding result concerning linear mappings from one vector space of symmetric matrices to another which preserve rank less than or equal to one. We also discuss some consequences of this characterization theorem.     

Sequences of vectors that are orbits of operators

This paper characterizes sequences of vectors that are the orbits of a linear operator and sequences of vectors in a Hilbert space that are orbits of a unitary operator. The latter is applied to time series. Sequences of vectors in a Hilbert space that generalize random walks are also shown to be the orbits of a bounded linear operator.     

A decomposable Hilbert space operator which is not strongly decomposable

In 1978 E. Albrecht constructed a Banach space operator which is decomposable in the sense of C. Foias, but not strongly decomposable. In the present note we describe a method which allows to construct examples of this kind in a systematic and much simpler way. In particular, we exhibit a decomposable, but not strongly decomposable operator on a Hilbert space and thus answer a corresponding question of M.Radjabalipour.