An observable on a quantum structure is any
σ-homomorphism of quantum structures from the Borel
σ-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form
(−∞,t) is sufficient to determine uniquely the whole observable defined on quantum structures like
σ-MV-algebras,
σ-effect algebras, Boolean
σ-algebras, monotone
σ-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.
We start with a heuristic description of a quantum mechanical system which admits ‘position’ as an observable in the sense of Definition 1.2.1. Suppose the position of the system can be identified with a point x in an abstract space X with the Borel structure B
B . We can then speak of the event ‘position x ϵ E’ or ‘x ϵ E’ for any E ϵ B
B . Suppose that ‘all’ the events concerning the system are described by the lattice P
P (H)
(H) of all orthogonal projections on a Hilbert space H
H . Then the event ‘x ϵ E’ can be identified with an element P(E) ϵ P
P (H)
(H) . Then the existence of a position observable is equivalent to the fact that the map E → P(E) from B
B to P
P (H)
(H) is a spectral measure.
Abstract Spectral automorphisms have been introduced in [IVANOV, A.—CARAGHEORGHEOPOL, D.: Spectral automorphisms in quantum logics, Internat. J. Theoret. Phys. 49 (2010), 3146–3152]_in an attempt to construct, in the abstract framework of orthomodular lattices, an analogue of the spectral theory in Hilbert spaces. We generalize spectral automorphisms to the framework of effect algebras with compression bases and study their properties. Characterizations of spectral automorphisms as well as necessary conditions for an automorphism to be spectral are given. An example of a spectral automorphism on the standard effect algebra of a finite-dimensional Hilbert space is discussed and the consequences of spectrality of an automorphism for the unitary Hilbert space operator that generates it are shown. The last section is devoted to spectral families of automorphisms and their properties, culminating with the formulation and proof of a Stone type theorem (in the sense of Stone’s theorem on strongly continuous one-parameter unitary groups — see, e.g. [REED, M.#x2014;SIMON, B.: Methods of Modern Mathematical Physics, Vol. I, Acad. Press, New York, 1975]) for a group of spectral automorphisms.
In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra
B X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on
B X constitute a convex set whose extreme points are degenerate probability measures.
Abstract In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers
p<q do the equalities
Tk=∫RxkF(dx) ,
k=p,q , imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if
p is odd and
q is even, and negative otherwise. The case
(p,q)=(1,2) closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on
C∗ -algebras is also provided.