Spectral Theory in Effect Algebras

Abstract

If one supposes a quantum logic L to be a σ-effect algebra, then the observables on L are identified with the L-valued measures defined on the Borel subsets of the real line. In this structure (and without the aid of Hibert space formalism) we will show that (1) the spectrum of an observable can be completely characterised by studying the observable (A - λ)-1, and (2) corresponding to every observable A there is a spectral resolution uniquely determined by A and uniquely determining A. Also, we study the existence of spectral measures corresponding to elements of a σ-MV-algebra, and we apply such a result to obtain a similar result concerning σ-complete lattice effect algebras.