Smearing of Observables and Spectral Measures on Quantum Structures

Abstract

Abstract. An observable on a quantum structure is any σ-homomorphism of quantum structures from the Borel σ-algebra of the real line into the quantum structure which is in our case a monotone σ-complete effect algebras with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean σ-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there is its spectral measure.

Remarks on effect-tribes

“…Therefore we also have a = h( a ∧ e) ⊕ h( a ∧ e ). We also know (see the proof of Theorem 3.4 in [4]) that for central elements e ∈ B 0 (M ) we have s(e) ∈ {0, 1} for all s ∈ ∂ e S(M ). Now a ∧ e(s) = s(a ∧ e), but s(a) = s(a ∧ e) + s(a ∧ e ).…”