Projections in a Synaptic Algebra |
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Authors: | David J Foulis Sylvia Pulmannová |
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Institution: | 1.Department of Mathematics and Statistics,University of Massachusetts,Amherst,USA;2.Mathematical Institute,Slovak Academy of Sciences,Bratislava,Slovakia |
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Abstract: | A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a
Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra
and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter
family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection
lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra
of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed
elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements
to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice,
and give several sufficient conditions for modularity of the projection lattice. |
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