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Spectral resolutions in Dedekind \(\sigma\)-complete \(\ell\)-groups. (English) Zbl 1072.06014
Summary: Using recent results on a generalized form of the Loomis-Sikorski theorem [A. Dvurečenskij, J. Aust. Math. Soc., Ser. A 68, 261–277 (2000; Zbl 0958.06006); D. Mundici, Adv. Appl. Math. 22, 227–248 (1999; Zbl 0926.06004)], it is shown that a unital Dedekind \(\sigma\)-complete \(\ell\)-group is a compatible Rickart comgroup in the sense of D. J. Foulis [Rep. Math. Phys. 54, 229–250 (2004)]. In particular, elements in unital Dedekind \(\sigma\)-complete \(\ell\)-groups and, consequently, elements in \(\sigma\)-MV-algebras, admit uniquely defined spectral resolutions similar to spectral resolutions of self-adjoint operators. A functional calculus and spectra of elements are considered in relation with the Loomis-Sikorski representation by functions.

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06D35 MV-algebras
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