Spectral resolutions in Dedekind σ-complete ℓ-groups

Abstract

Using recent results on a generalized form of the Loomis–Sikorski theorem [A. Dvurečenskij, Loomis–Sikorski theorem for σ-complete MV-algebras and ℓ-groups, J. Austral. Math. Soc. Ser. A 68 (2000) 261–277; D. Mundici, Tensor product and the Loomis–Sikorski theorem for MV-algebras, Adv. Appl. Math. 22 (1999) 227–248], it is shown that a unital Dedekind σ-complete ℓ-group is a compatible Rickart comgroup in the sense of D.J. Foulis [D.J. Foulis, Spectral resolutions in a Rickart comgroup, Rep. Math. Phys. 54 (2004) 229–250]. In particular, elements in unital Dedekind σ-complete ℓ-groups and, consequently, elements in σ-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.