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 (Int. J. Theor. Phys. 49(12):3146–3152, 2010) to create an analogue of the Hilbert space spectral theory in the abstract framework of orthomodular lattices. We develop the theory of spectral automorphisms finding 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 automorphism 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 analogue of the Stone theorem concerning strongly continuous one-parameter unitary groups is given.