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A projection and an effect in a synaptic algebra. (English) Zbl 1357.17030

Summary: We study a pair \(p, e\) consisting of a projection \(p\) (an idempotent) and an effect \(e\) (an element between 0 and 1) in a synaptic algebra (a generalization of the self-adjoint part of a von Neumann algebra). We show that some of Halmos’s theory of two projections (or two subspaces), including a version of his CS-decomposition theorem, applies in this setting, and we introduce and study two candidates for a commutator projection for \(p\) and \(e\).

MSC:

17C65 Jordan structures on Banach spaces and algebras
06C15 Complemented lattices, orthocomplemented lattices and posets
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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References:

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