Chapter 21 - Probability on MV-Algebras

Abstract

This chapter surveys the MV-algebraic generalization of notions and results of Boolean algebraic probability theory. MV-algebras are a noncommutative generalization of Boolean algebras. In the approach of Birkhoff and von Neumann, properties of a quantum system measure are identified with self-adjoint idempotent elements in the algebra of continuous Hilbert operators on the Hilbert space. No Hilbert space can be canonically assigned to a system with infinitely many degrees of freedom, such as those naturally occurring in quantum statistical mechanics and in quantum field theory. The MV-algebraic probability theory developed in this survey can be thought of as a noncommutative generalization of classical Boolean algebraic probability theory. An MV-algebra is said to be semisimple if for each nonzero element, there is a homomorphism. The weak σ-distributivity of the underlying group structure of any Riesz space is a necessary and sufficient condition for every valued measure to be extendable from the subalgebra of a set to its enveloping σ-algebra. The MV-algebraic central limit theorem is also elaborated in the chapter.