Norm-attaining functionals onC(Q, X)

Functionals (vector measures) defined on the spaceC(Q, X) of continuous abstract functions (whereQ is a compact Hausdorff space andX is a Banach space) and attaining their norm on the unit sphere are considered. A characterization of such functionals is given in terms of the Radon-Nikodym derivative of the vector measure with respect to the variation of the measure and in terms of analogs of the derivative. Applications to the characterization of finite-codimensional subspaces with the best approximation property are given. Similar results are obtained for the spaceB(Q, Σ, X) of uniform limits of simple functions. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 45–56, January, 1997. Translated by V. E. Nazaikinskii     

The algebraic structure ofH-dissipative operators in a finite-dimensional space

We study properties of Jordan representations ofH-dissipative operators in a finite-dimensional indefiniteH-space. An algebraic proof is given of the fact that such operators always have maximal semidefinite invariant subspaces. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 163–169, February, 1998. The authors axe grateful to Professor A. A. Shkalikov for useful discussions. The research of the first author was supported by INTAS under grant No. 93-0249. The research of the second author was supported by the International Science Foundation and the Russian Government under grant No. NZP300.     

The Maslov Gerbe

Let Lag(E) be the Grassmannian of Lagrangian subspaces of a complex symplectic vector space E. We construct a Maslov class which generates the second integral cohomology of Lag(E), and we show that its mod 2 reduction is the characteristic class of a flat gerbe with structure group Z ₂. We explain the relation of this gerbe to the well-known flat Maslov line bundle with structure group Z ₄ over the real Lagrangian Grassmannian, whose characteristic class is the mod 4 reduction of the real Maslov class.     

Subordination,rank, and determinism of multivariate stationary sequences

Necessary and sufficient conditions for an arbitrary q-variate stationary sequence x_t, t ∈ $\text{Z}$, to be deterministic are presented. A characterization of the rank r(x) of x_t, t ∈ $\text{Z}$, and a method to construct the Wold-Cramér decomposition for x_t, t ∈ $\text{Z}$, are given. Subordination of q-variate bounded orthogonally scattered vector measures is considered.     

The illumination conjecture and its extensions

Summary The Illumination Conjecture was raised independently by Boltyanski and Hadwiger in 1960. According to this conjecture any <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>d$-dimensional convex body can be illuminated by at most $2^d$ light sources. This is an important fundamental problem. The paper surveys the state of the art of the Illumination Conjecture.     

Polycyclic codes as invariant subspaces

Polycyclic codes are a powerful generalization of cyclic and constacyclic codes. Their algebraic structure is studied here by the theory of invariant subspaces from linear algebra. As an application, a bound on the minimum distance of these codes is derived which outperforms, in some cases, the natural analogue of the BCH bound.     

因子von Neumann代数上导子的等价刻画

设R是含非平凡幂等元P的素环,C∈R,C=PC.本文证明可加映射△:R→R在C可导,即△(AB)=△(A)B+A△(B),A,B∈R,AB=C当且仅当存在导子δ:R→R,使得△(A)=δ(A)+△(I)A,A∈R.没有I_1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子C∈B(X),使得ran(C)或ker(C)在X中可补,则可加映射△:B(X)→B(X)在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.     

Dual subspaces of operators

In this note we introduce some new constructions of dual spaces of operators, which are, of course, at the same time, operator spaces in the sense of Pisier (2003) [12]. We exemplify the utility of these constructs by establishing, in this more general setting, a curious and little known result from the theory of dual algebras, namely from Chevreau and Pearcy (1991) [11].     

Invariant subspaces of X** under the action of biconjugates

We study conditions on an infinite dimensional separable Banach space X implying that X is the only non-trivial invariant subspace of X** under the action of the algebra of biconjugates of bounded operators on . Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of c ₀, and show in particular that any space which does not contain ℓ₁ and has property (u) of Pelczynski is simple.     

Affiliated subspaces and infiniteness of von Neumann algebras

We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi‐splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi‐splitting subspaces are non‐equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi‐splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.