A new class of Kadison-Singer algebras

We show that the projection lattice generated by a maximal nest and a rank one projection in a separable infinite-dimensional Hilbert space is in general reflexive. Moreover we show that the corresponding reflexive algebra has a maximal triangular property, equivalently, it is a Kadison-Singer algebra. Similar results are also obtained for the lattice generated by a finite nest and a projection in a finite factor.     

Volume and Lattice Points of Reflexive Simplices

Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected to contain the largest number of lattice points even among all lattice polytopes with only one interior lattice point. Translated in algebro-geometric language, our main theorem yields a sharp upper bound on the anticanonical degree of d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number one, e.g., of weighted projective spaces with Gorenstein singularities.     

Minimal generating reflexive lattices of projections in finite von Neumann algebras

We show that the reflexive lattice generated by a double triangle lattice of projections in a finite von Neumann algebra is topologically homeomorphic to the two-dimensional sphere S ² (plus two distinct points corresponding to zero and I). Furthermore, such a reflexive lattice is in general minimally generating for the von Neumann algebra it generates. As an application, we show that if a reflexive lattice F{\mathcal F} generates the algebra M_n(\mathbb C){M_n(\mathbb C)} of all n × n complex matrices, for some n ≥ 3, then F\{0,I}{\mathcal F\setminus\{0,I\}} is connected if and only if it is homeomorphic to S ².     

On some maximal non-selfadjoint operator algebras

We show that the reflexive algebra given by the lattice generated by a maximal nest and a rank one projection is maximal with respect to its diagonal.     

Pretopologies, preuniformities and probabilistic metric spaces

Summary A pretopology on a given set can be generated from a filter of reflexive relations on that set (we call such a structure a preuniformity). We show that the familly of filters inducing a given pretopology on Xform a complete lattice in the lattice of filters on X. The smallest and largest elements of that lattice are explicitly given. The largest element is characterized by a condition which is formally equivalent to a property introduced by Knaster--Kuratowski--Mazurkiewicz in their well known proof of Brouwer's fixed point theorem. Menger spaces and probabilistic metric spaces also generate pretopologies. Semi-uniformities and pretopologies associated to a possibly nonseparated Menger space are completely characterized.     

Notes on the Roots of Ehrhart Polynomials

We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n², where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n². For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes. We further give a characterisation of the roots of Ehrhart polyomials in the three-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.     

Representations admitting two pairs of supplementary invariant spaces

We examine the lattice generated by two pairs of supplementary vector subspaces of a finite-dimensional vector-space by intersection and sum, with the aim of applying the results to the study of representations admitting two pairs of supplementary invariant spaces, or one pair and a reflexive form. We show that such a representation is a direct sum of three canonical sub-representations which we characterize. We then focus on representations of Berger algebras with the same property.     

Commutants modulo the compact operators of certain CSL algebras II

The commutant modulo compacts, or essential commutant, of a reflexive algebra with commutative subspace lattice is a C^* algebra which is the sum of the compact operators in L(H) and a C^* subalgebra of the core. We give a characterization of the essential commutant of a separably acting CSL algebra in terms of properties of the spectral measure of an operator in the intersection of the essential commutant and the core. This is used to determine some sufficient conditions on the lattice for when the essential commutant is norm generated by the projections it contains.     

Rank-one operators in reflexive one-sided<Emphasis Type="Italic">A</Emphasis>-submodules

In this paper, we first characterize reflexive one-sided A-submodulesU of a unital operator algebraA inB(H) completely. Furthermore we investigate the invariant subspace lattice LatR and the reflexive hull RefR, whereR is the submodule generated by rank-one operators inU; in particular, ifL is a subspace lattice, we obtain when the rank-one algebraR of AlgL is big enough to determined AlgL in the following senses: AlgL = Alg LatR and AlgL = RefR.     

弱L^p的对偶空间的一个注记

本文证明了非自反弱序列完备Ｂａｎａｃｈ格在它的二次对偶空间中的正交补是非自反映的，从而证明了Ｃｗｉｋｅｌ的关于弱Ｌ＾ｐ的对偶空间的奇异部的表示定理是错误的。     

Ehrhart Series and Lattice Triangulations

We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms of multivariate analogues of the h-polynomials of the subdivision and “local contributions” of the links of its nonunimodular faces. We also compute new examples of nonunimodal h ^*-vectors of reflexive polytopes. Supported by the Clay Mathematics Institute.     

Fixed point theorems for nonexpansive mappings and Suzuki-generalized nonexpansive mappings on a Banach lattice

We prove the existence of fixed points for multivalued nonexpansive nonself-mappings on a weakly orthogonal reflexive Banach lattice with uniformly monotone norm. Moreover, for single-valued mappings, we extend Betiuk-Pilarska and Prus’s result [A. Betiuk-Pilarska, S. Prus, Banach lattices which are order uniformly noncreasy, J. Math. Anal. Appl. 342 (2008) 1271–1279] on the weak fixed point property to continuous mappings satisfying condition (C) on a w-weakly orthogonal OUNC Banach lattice.     

Convergence of Bregman Projection Methods for Solving Consistent Convex Feasibility Problems in Reflexive Banach Spaces

The problem that we consider is whether or under what conditions sequences generated in reflexive Banach spaces by cyclic Bregman projections on finitely many closed convex subsets Q _i with nonempty intersection converge to common points of the given sets.     

自反算子代数的模和交换子以及一阶上同调空间

设L是赋范线性空间上的子空间格,一个子空间是自反AlgL-模的充分必要条件被得到,当L是完全分配子空间格时,自反AlgL-模的二次交换子被描述,进而,本文引入V-生成子稠格,这是一种严格地包含了完全分配格和五角格的格类。当L是可换的V-生成子稠格时,模模交换子C(AlgL;M)和代数AlgLatM都被分解成直和,并且满足条件H~1(AlgL,B(H))=0的一阶上同调空间H~1(AlgL,M)被刻划。     

Unimodality Questions for Integrally Closed Lattice Polytopes

It is a famous open question whether every integrally closed reflexive polytope has a unimodal Ehrhart δ -vector. We generalize this question to arbitrary integrally closed lattice polytopes and we prove unimodality for the δ -vector of lattice parallelepipeds. This is the first nontrivial class of integrally closed polytopes. Moreover, we suggest a new approach to the problem for reflexive polytopes via triangulations.     

Reflexivity of bilattices

We study reflexivity of bilattices. Some examples of reflexive and non-reflexive bilattices are given. With a given subspace lattice ? we may associate a bilattice Σ_?. Similarly, having a bilattice Σ we may construct a subspace lattice Σ_?. Connections between reflexivity of subspace lattices and associated bilattices are investigated. It is also shown that the direct sum of any two bilattices is never reflexive.     

Local derivations and local automorphisms

In this paper, we show that if is a completely distributive commutative subspace lattice or a -subspace lattice, then the space of all bounded derivations of is reflexive. We also study when local automorphisms on some algebras are automorphisms.     

Lazarsfeld-Mukai reflexive sheaves and their stability

Consider an ample and globally generated line bundle L on a smooth projective variety X of dimension N≥2 over ?. Let D be a smooth divisor in the complete linear system of L. We construct reflexive sheaves on X by an elementary transformation of a trivial bundle on X along certain globally generated torsion-free sheaves on D. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the μ_L-(semi)stability of such reflexive sheaves under certain conditions.     

Banach Lattices on Which Every Power-Bounded Operator is Mean Ergodic

Given a Banach lattice E that fails to be countably order complete, we construct a positive compact operator A: E E for which T = I - A is power-bounded and not mean ergodic. As a consequence, by using the theorem of R. Zaharopol, we obtain that if every power-bounded operator in a Banach lattice is mean ergodic then the Banach lattice is reflexive.     

Reflexive cones

Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of $\ell _{1}$ . Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the spaces with the Schur property by the properties of reflexive cones is given.