Characterizations of spectral automorphisms and a Stone-type theorem in orthomodular lattices

Abstract

The notion of spectral automorphism of an orthomodular lattice was introduced by Ivanov and Caragheorgheopol [3] to create an analogue of the Hilbert space spectral theory in the abstract framework of orthomodular lattices. We develop the theory of spec- tral automorphisms nding previously missing characterizations of spectral automorphisms, discussing several examples and the possibility to construct such automorphisms in direct products or horizontal sums of lattices. A factorization of the spectrum of a spectral auto- morphism is found. The last part of the paper addresses the problem of the unitary time evolution of a system from the point of view of the spectral automorphisms theory. An ana- logue of the Stone theorem concerning strongly continuous one-parameter unitary groups is given.