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Spectral resolution in a Rickart comgroup. (English) Zbl 1161.81310
Summary: A comgroup is a compressible group with the general comparability property. A comgroupwith the Rickart projection property is called a Rickart comgroup. We show that each element of a Rickart comgroup has a rational spectral resolution and a nonempty closed and bounded (real) spectrum. The rational spectral resolution and the spectrum are shown to have many of the properties of the spectral resolution and spectrum of a self-adjoint operator on a Hilbert space. Examples of Rickart comgroups include the additive group of self-adjoint elements in a von Neumann algebra and the Mundici group of a Heyting MV algebra.

MSC:
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
47B25 Linear symmetric and selfadjoint operators (unbounded)
47N50 Applications of operator theory in the physical sciences
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