synaptic algebra
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2018 ◽  
Vol 51 (1) ◽  
pp. 1-7 ◽  
Author(s):  
David J. Foulis ◽  
Sylvia Pulmannová

Abstract We prove that if A is a synaptic algebra and the orthomodular lattice P of projections in A is complete, then A is a factor if and only if A is an antilattice.We also generalize several other results of R. Kadison pertaining to infima and suprema in operator algebras.



Order ◽  
2018 ◽  
Vol 36 (1) ◽  
pp. 1-17 ◽  
Author(s):  
David J. Foulis ◽  
Sylvia Pulmannová


2017 ◽  
Vol 67 (6) ◽  
Author(s):  
David J. Foulis ◽  
Anna Jenčová ◽  
Sylvia Pulmannová

AbstractA synaptic algebra



Positivity ◽  
2016 ◽  
Vol 21 (3) ◽  
pp. 919-930 ◽  
Author(s):  
David J. Foulis ◽  
Anna Jenčová ◽  
Sylvia Pulmannová
Keyword(s):  


2016 ◽  
Vol 66 (2) ◽  
Author(s):  
David J. Foulis ◽  
Sylvia Pulmannová

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.



2015 ◽  
Vol 485 ◽  
pp. 417-441 ◽  
Author(s):  
David J. Foulis ◽  
Anna Jenčová ◽  
Sylvia Pulmannová
Keyword(s):  


2015 ◽  
Vol 478 ◽  
pp. 162-187 ◽  
Author(s):  
David J. Foulis ◽  
Anna Jenčová ◽  
Sylvia Pulmannová
Keyword(s):  


2014 ◽  
Vol 64 (3) ◽  
Author(s):  
David Foulis ◽  
Sylvia Pulmannová

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.



2013 ◽  
Vol 43 (8) ◽  
pp. 948-968 ◽  
Author(s):  
David J. Foulis ◽  
Sylvia Pulmannová


2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Sylvia Pulmannová

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.



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