On States on MV-algebras and their Applications
Anatolij Dvurečenskij · Journal of Logic and Computation · 2009
In the frames of quantum structures, states are a very important notion that model probability on an algebraic structure. Already G. Boole observed that to calculate probability of event of the structure M, it is important to know only which two events A and B we can add such that if C = A+B, and P is a probability, then P (A+B) = P (A) + P (B); and the operation + is a partial one on M. If M is a Boolean algebra, then A+B: = A∪B whenever A∩B = ∅ or equivalently, A ≤ B′. Such two events A and B are said to be mutually excluding or summable, [DvPu]. Therefore, the state or finitely additive state on an algebraic structure (M; +, ′ , 0, 1) is any mapping s: M → [0, 1] such that (i) s(1) = 1, and (ii) s(a + b) = s(a) + s(b) whenever a + b is defined in M. The unary operation ′ is an orthogonal complement or a negation. The basic task is how to define a state on an algebraic structure like (M;⊕,, ∗ , 0, 1) or (M;⊕,, −, ∼ , 0, 1), that is, how to derive a partial operation + from the ⊕,? In more complicated structures, like orthomodular lattices and effect algebras, the most important example is the system L(H) of all closed subspaces of a Hilbert space H or the system of all Hermitian operators E(H) that are between the zero operator and the identity operator. Here the σ-additive states are of the form sφ(M) = (PMφ, φ),M ∈ L(H), φ ∈ H, s(M) = i λisφ(M) = tr(TPM), M ∈ L(H).