scholarly journals Symmetries in synaptic algebras

2014 ◽  
Vol 64 (3) ◽  
Author(s):  
David Foulis ◽  
Sylvia Pulmannová
Keyword(s):  
Jordan Algebra ◽  
Equivalence Relation ◽  
Orthomodular Lattice ◽  
Von Neumann Algebra ◽  
Unit Element ◽  
Von Neumann ◽  
Synaptic Algebra ◽  
Projection Lattice ◽  
Finite Sequences

AbstractA synaptic algebra is a generalization of the Jordan algebra of self-adjoint elements of a von Neumann algebra. We study symmetries in synaptic algebras, i.e., elements whose square is the unit element, and we investigate the equivalence relation on the projection lattice of the algebra induced by finite sequences of symmetries. In case the projection lattice is complete, or even centrally orthocomplete, this equivalence relation is shown to possess many of the properties of a dimension equivalence relation on an orthomodular lattice.

Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups

2018 ◽  
Vol 30 (4) ◽  
pp. 973-995 ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Orthomodular Lattice ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Commutative Group ◽  
Von Neumann ◽  
Natural Embedding ◽  
New Class ◽  
Projection Lattice ◽  
Archimedean Property ◽  
Initial Object

AbstractIt is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular latticeX, admits a natural embedding into a group{G(X)}with a lattice ordering so that{G(X)}determinesXup to isomorphism. The embedding{X\hookrightarrow G(X)}appears to be a universal (non-commutative) group-valued measure onX, while states ofXturn into real-valued group homomorphisms on{G(X)}. The existence of completions is characterized by a generalized archimedean property which simultaneously applies toXand{G(X)}. By an extension of Foulis’ coordinatization theorem, the negative cone of{G(X)}is shown to be the initial object among generalized Baer{{}^{\ast}}-semigroups. For finiteX, the correspondence betweenXand{G(X)}provides a new class of Garside groups.

Commutativity in a synaptic algebra

2016 ◽  
Vol 66 (2) ◽  
Author(s):  
David J. Foulis ◽  
Sylvia Pulmannová
Keyword(s):  
Von Neumann Algebra ◽  
Vector Lattice ◽  
The Self ◽  
Natural Conditions ◽  
Von Neumann ◽  
Synaptic Algebra ◽  
Functional Representation

AbstractA synaptic algebra is a generalization of the self-adjoint part of a von Neumann algebra. For a synaptic algebra we study two weakened versions of commutativity, namely quasi-commutativity and operator commutativity, and we give natural conditions on the synaptic algebra so that each of these conditions is equivalent to commutativity. We also investigate the structure of a commutative synaptic algebra, prove that a synaptic algebra is commutative if and only if it is a vector lattice, and provide a functional representation for a commutative synaptic algebra.

A note on ideals in synaptic algebras

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Sylvia Pulmannová
Keyword(s):  
Jordan Algebra ◽  
Normed Space ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Algebra Structure ◽  
Order Unit ◽  
Synaptic Algebra ◽  
Special Jordan Algebra

AbstractThe notion of a synaptic algebra was introduced by David Foulis. Synaptic algebras unite the notions of an order-unit normed space, a special Jordan algebra, a convex effect algebra and an orthomodular lattice. In this note we study quadratic ideals in synaptic algebras which reflect its Jordan algebra structure. We show that projections contained in a quadratic ideal from a p-ideal in the orthomodular lattice of projections in the synaptic algebra and we find a characterization of those quadratic ideals which are generated by their projections.

Reflexivity for a class of subspace lattices

1996 ◽  
Vol 119 (1) ◽  
pp. 67-71
Author(s):  
E. G. Katsoulis
Keyword(s):  
Complete Lattice ◽  
Von Neumann Algebra ◽  
Von Neumann ◽  
Projection Lattice ◽  
Closed Lattice ◽  
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.

Synaptic algebras

2010 ◽  
Vol 60 (5) ◽  
Author(s):  
David Foulis
Keyword(s):  
Hilbert Space ◽  
Jordan Algebra ◽  
Normed Space ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Order Unit ◽  
Natural Conditions ◽  
Spectral Order ◽  
Synaptic Algebra ◽  
Special Jordan Algebra

AbstractA synaptic algebra is both a special Jordan algebra and a spectral order-unit normed space satisfying certain natural conditions suggested by the partially ordered Jordan algebra of bounded Hermitian operators on a Hilbert space. The adjective “synaptic”, borrowed from biology, is meant to suggest that such an algebra coherently “ties together” the notions of a Jordan algebra, a spectral order-unit normed space, a convex effect algebra, and an orthomodular lattice.

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

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