Metrics on Projections of the Von Neumann Algebra Associated with Tracial Functionals

Siberian Mathematical Journal - Let φ be a positive functional on a von Neumann algebra $$\mathscr{A}$$ and let $$\mathscr{A}^{\rm{pr}}$$ be the projection lattice in $$\mathscr{A}$$. Given...     

On some vector lattices of operators and their finite elements

If the vector space of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for to be a vector lattice. and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space or, in the case that is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices , where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in . The paper contains also some generalization of results obtained for the case in [10].      

Matroid bases with cardinality constraints on the intersection

Mathematical Programming - Given two matroids $$\mathcal {M}_{1} = (E, \mathcal {B}_{1})$$ and $$\mathcal {M}_{2} = (E, \mathcal {B}_{2})$$ on a common ground set E with base sets $$\mathcal...     

On lattices of convex sets in {\mathbb{R}}^{n}

Properties of several sorts of lattices of convex subsets of are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of and the lattice of all convex subsets of The lattices of arbitrary, of open bounded, and of compact convex sets in all satisfy the same identities, but the last of these is join-semidistributive, while for n > 1 the first two are not. The lattice of relatively convex subsets of a fixed set satisfies some, but in general not all of the identities of the lattice of “genuine” convex subsets of To the memory of Ivan RivalReceived April 22, 2003; accepted in final form February 16, 2005.This revised version was published online in August 2005 with a corrected cover date.     

Projectivity of the free Banach lattice generated by a lattice

We study the projectivity of the free Banach lattice generated by a lattice $${\mathbb {L}}$$ in two cases: when the lattice is finite, and when the lattice is an infinite linearly ordered set. We prove that in the first case, it is projective while in the second case, it is not.     

Rings of Integers in Number Fields and Root Lattices

Doklady Mathematics - This paper investigates whether a root lattice can be similar to the lattice $$\mathcal{O}$$ of all integer elements of a number field K endowed with the inner product...     

A characterization of complex lattice homomorphisms

In this paper we discuss the problem of how to recognize a complex lattice homomorphism on the complexification of a real vector lattice L from its behavior on a small subset of .      

Characterization of symmetrically $$\Delta $$ Δ -normed operator ideals which are interpolation spaces between Schatten–von Neumann ideals

Positivity - Let $$\mathcal {E}$$ be a symmetrically $$\Delta $$ -normed ideal in B(H). For $$1\le p<\infty ,$$ $$q\ge 1,$$ we give a necessary and sufficient condition for $$\mathcal {E}$$...     

Sums, products and negations of contexts and complete lattices

In Formal Concept Analysis, one associates with every context its concept lattice , and conversely, with any complete lattice L the standard context L, constituted by the join-irreducible elements as ‘objects’, the meet-irreducible elements as ‘attributes’, and the incidence relation induced by the lattice order. We investigate the effect of the operators and on various (finite or infinite) sum and product constructions. The rules obtained confirm the ‘exponential’ behavior of and the ‘logarithmic’ behavior of with respect to cardinal operations but show a ‘linear’ behavior on ordinal sums. We use these results in order to establish several forms of De Morgan’s law for the lattice-theoretical negation operator, associating with any complete lattice the concept lattice of the complementary standard context. Received February 7, 2001; accepted in final form January 6, 2006.     

Transitive quasi-uniform structures depending on a parameter

Aequationes mathematicae - In a category $${\mathcal {C}}$$ with an ( $${\mathcal {E}}$$ , $${\mathcal {M}}$$ )-factorization structure for morphisms, we prove that any subclass $${\mathcal {N}}$$...     

The Topology on the Space of δ-psh Functions in the Cegrell Classes

The aim of the present paper is to introduce a metric locally convex topology on the space of δ-psh functions in the Cegrell class . We prove that with this topology is a non-separable and non-reflexive Fréchet space. At the same time, we extend the Monge–Ampère operator from the class to .     

Group Pairs with Property (T), from Arithmetic Lattices

Let Γ be an arithmetic lattice in an absolutely simple Lie group G with trivial centre. We prove that there exists an integer N ≥ 2, a subgroup Λ of finite index in Γ, and an action of Λ on such that the pair ( ) has property (T). If G has property (T), then so does . If G is the adjoint group of Sp(n, 1), then is a property (T) group satisfying the Baum–Connes conjecture. If Γ is an arithmetic lattice in SO(n, 1), then the associated von Neumann algebra is a II₁-factor in Popa’s class . Elaborating on this result of Popa, we construct a countable family of pairwise nonstably isomorphic group II₁-factors in the class , all with trivial fundamental groups and with all L²-Betti numbers being zero.Mathematics Subject Classiffications (2000). 22E40, 22E47, 46L80, 37A20     

Fields of quotients of lattice-ordered domains

It is not known whether the field of fractions of an integral domain with a compatible lattice order has a compatible lattice order that extends the given order on the integral domain. The polynomial ring over the real numbers has a natural compatible lattice order, viz, the coordinatewise order . We describe circumstances in which the field of fractions of has no archimedean lattice order that extends .Received May 2, 2003; accepted in final form June 4, 2004.     

Generalization of the theorems of Barndorff-Nielsen and Balakrishnan–Stepanov to Riesz spaces

Positivity - In a Dedekind complete Riesz space, E, we show that if $$(P_n)$$ is a sequence of band projections in E then $$\begin{aligned} \limsup \limits _{n\rightarrow \infty } P_n - \liminf...     

On a class of real normed lattices

We say that a real normed lattice is quasi-Baire if the intersection of each sequence of monotonic open dense sets is dense. An example of a Baire-convex space, due to M. Valdivia, which is not quasi-Baire is given. We obtain that E is a quasi-Baire space iff is a pairwise Baire bitopological space, where , is a quasi-uniformity that determines, in L. Nachbin's sense, the topological ordered space E.     

The Lattice of Cyclic Flats of a Matroid

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice of sets and a function to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to . We apply this perspective to give an alternative view of the free product of matroids and we show how to compute the Tutte polynomial of the free product in terms of the Tutte polynomials of the constituent matroids. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids. Received May 29, 2005     

Join congruence relations and closure relations

In this paper we show that, given a complete lattice , the following three lattices are the same: (1) the lattice of closure relations on , (2) the lattice of meet-closed subsets of , and (3) the lattice of complete join congruence relations on .     

Optimal Extensions for pth Power Factorable Operators

Mediterranean Journal of Mathematics - Let $${X(\mu)}$$ be a function space related to a measure space $${(\Omega,\Sigma,\mu)}$$ with $${\chi_\Omega\in X(\mu)}$$ and let $${T\colon X(\mu)\to E}$$...