Perspectivity in the Projection Lattice of an AW ∗ -Algebra

1973 ◽  
Vol 38 (2) ◽  
pp. 367
Author(s):  
George A. Elliott
Keyword(s):  
Projection Lattice

Regular elements in generalized hermitian algebras

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
David Foulis ◽  
Sylvia Pulmannová
Keyword(s):  
Spectral Order ◽  
Type Representation ◽  
Von Neumann ◽  
Regular Elements ◽  
Unit Space ◽  
Special Jordan Algebra ◽  
Projection Lattice ◽  
Von Neumann Regular ◽  
Algebra Relate ◽  
Order Unit Space

AbstractA generalized Hermitian (GH) algebra is a special Jordan algebra that is at the same time a spectral order-unit space. In this paper we characterize the von Neumann regular elements in a GH-algebra, relate maximal pairwise commuting subsets of the algebra to blocks in its projection lattice, and prove a Gelfand-Naimark type representation theorem for commutative GH-algebras.

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.

scholarly journals The order topology on the projection lattice of a Hilbert space

2012 ◽  
Vol 159 (9) ◽  
pp. 2280-2289 ◽  
Author(s):  
D. Buhagiar ◽  
E. Chetcuti ◽  
H. Weber
Keyword(s):  
Hilbert Space ◽  
Order Topology ◽  
Projection Lattice

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.

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.

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