Lattice Of Projections Research Articles

Reflexivity for a class of subspace lattices

AbstractThe complete lattice generated by a totally atomic CSL ℒ and the projection lattice of a von Neumann algebra ℛ, commuting with ℒ, is reflexive. From this it follows that the strongly closed lattice generated by any CSL ℒ and the projection lattice of a properly infinite von Neumann algebra ℛ, commuting with ℒ, is reflexive.

Projective mappings between projective lattice geometries

The concept of projective lattice geometry generalizes the classical synthetic concept of projective geometry, including projective geometry of modules.

Faux-boolean algebras and classical models

A certain substructure of the lattice of projections on a Hilbert space is defined, and it is shown under what conditions such a structure can be said to have a “classical model” (in a sense made precise). The results have implications for the interpretation of quantum mechanics.

Orthosymmetric ortholattices and Rickart*-rings

Orthosymmetric ortholattices (OSOLs) have been introduced in order to approximate ortholattices of closed subspaces of a Hilbert space. In this paper, some new properties of OSOLs are proved and the main result states that lattices of projections of Rickart*-rings, satisfying 2x=0 impliesx=0, carry a natural structure of OSOL.

Homology for Operator Algebras I. Spectral Homology for Reflexive Algebras

A stable homology theory is defined for completely distributive CSL algebras in terms of the point-neighbourhood homology of the partially ordered set of meet-irreducible elements of the invariant projection lattice. This specialises to the simplicial homology of the underlying simplicial complex in the case of a digraph algebra. These groups are computable and useful. In particular it is shown that if the first spectral homology group is trivial then Schur automorphisms are automatically quasispatial. This motivates the introduction of essential Hochschild cohomology which we define by using the point weak star closure of coboundaries in place of the usual coboundaries.

Traces and subadditive measures on projections in 𝐽𝐵𝑊-algebras and von Neumann algebras

Let P ( M ) P(M) be the projection lattice of an arbitrary JBW-algebra or von Neumann algebra M. It is shown that the tracial states of M correspond by extension precisely to the subadditive probability measures on P ( M ) P(M) . The analogous result for normal semifinite traces is also proved.

On subsemigroup lattices without non-trivial identities

A latticeL is called discriminating if for any free latticeF and for any finite number of elementsu1,u2, ...,un∈F, there exists a homomorphismf:F→L such thatf(ui)≠f(uj) wheneverui ≠ uj (1≤i, j≤n). In this paper it is proved that the subsemigroup lattice SubS of a commutative semigroupS does not satisfy a non-trivial identity if and only if SubS is discriminating. In particular, in this case every finite projective lattice can be embedded into SubS. It should be noted that the most important examples of semigroups whose subsemigroup lattices satisfy no non-trivial identity and therefore have the discriminating property are the following: the infinite cyclic semigroup, the free semilattice of countable rank, any commutative nilsemigroup which is not nilpotent and so on.

Constructive complete distributivity IV

A complete latticeL isconstructively completely distributive, (CCD), when the sup arrow from down-closed subobjects ofL toL has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and sup-preserving arrows. There is a restriction to order ideals and “totally algebraic” lattices. Both biequivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to yet another characterization of (CCD) lattices in a topos.

Measure on the inductive limit of projection lattices

A probability measure on a nondecreasing net of lattices of orthogonal projections in von Neumann algebras is extended to a probability on the inductive limit of the lattices.

States on projection logics of von Neumann algebras

We summarize recent results concerning states on projection lattices of von Neumann algebras. In particular, we present an analysis of the Jauch-Piron property in the von Neumann algebra setting.

Global-Affine Morphisms of Projective Lattice Geometries

The fundamental theorem of projective geometry gives an algebraic representation of isomorphisms between projective geometries of dimension at least 3 over vector spaces and has been generalized in different ways. This note briefly presents some further generalizations which will be proved in the author’s thesis. We introduce the notion of global-affine morphisms between projective lattice geometries. Our investigations result in a general partial representation of global-affine morphisms which yields a complete representation of global-affine homomorphisms between large classes of module-induced projective geometries by semilinear mappings between the underlying modules.

Boolean powers and quantum measurements

A description of a quantum measurement is proposed, where the measured system is described in the frame of the traditional Hilbert space approach and the measuring apparatus is represented by a complete Boolean algebra. The event structure of the coupled system is described by the Boolean power of the projection lattice of the Hilbert space with respect to the Boolean algebra.

Proportions of Different Unit-Cells in a Projection Lattice

A general formula for determinating the proportions of different unit-cells in a projection lattice is given. It is found that there exist certain relations between the occurrence frequency of a specified unit-cell in the aperiodic lattice and the projection (in a special subspace E2') of the corresponding facets of the hyper-unit-cell of the n-dimensional periodic lattice. Based upon these, some special cases are discussed.

A unified approach to projective lattice geometries

The interest in pursuing projective geometry on modules has led to several lattice theoretic generalizations of the classical synthetic concept of projective geometry on vector spaces.

On Barbilian spaces in projective lattice geometries

We introduce the notion of a Barbilian space of a projective lattice geometry in order to investigate the relationship between lattice-geometric properties and the properties of point-hyperplane structures associated with. We obtain a characterization of those projective lattice geometries, the Barbilian space of which is a Veldkamp space.

A Compact Lattice of Projections

A Compact Lattice of Projections

Lattices over integral group rings and abelian subquotients

The aim of this paper is to define and investigate the properties of an interesting class of lattices over an integral group ring, generalizing projective such lattices. This class includes any ring of algebraic integers considered as a Galois-module.

The Mackey-Gleason Problem

Let A be a von Neumann algebra with no direct summand of Type ${{\text {I}}_2}$, and let $\mathcal {P}(A)$ be its lattice of projections. Let X be a Banach space. Let $m:\mathcal {P}(A) \to X$ be a bounded function such that $m(p + q) = m(p) + m(q)$ whenever p and q are orthogonal projections. The main theorem states that m has a unique extension to a bounded linear operator from A to X. In particular, each bounded complex-valued finitely additive quantum measure on $\mathcal {P}(A)$ has a unique extension to a bounded linear functional on A.

Finite Lattices of Projections in Factors and Approximately Finite C*-Algebras

AbstractA unique factorisation theorem is obtained for tensor products of finite lattices of commuting projections in a factor. This leads to unique tensor product factorisations for reflexive subalgebras of the hyperfinite II1 factor which have irreducible finite commutative invariant projection lattices. It is shown that the finite refinement property fails for simple approximately finite C*-algebras, and this implies that there is no analogous general result for finite lattice subalgebras in this context.

Orthosymmetric ortholattices

Orthosymmetric ortholattices, which form an equational class of algebras, approximate ortholattices of closed subspaces of Hilbert spaces, and more generally projection lattices of von Neumann algebras, more closely than orthomodular lattices