Quantum structures and operator algebras

Abstract

This chapter summarizes both classical and recent results concerning the extension of fundamental principles of measure theory to the projection logics of operator algebras. It is noted that in standard measure theory, the basic concept is that of a measure on a σ-field, A, of subsets of a set Ω. A is a Boolean algebra with respect to the set theoretic operations and the corresponding set of A-measurable functions on Ω constitutes a commutative algebra with respect to the arithmetic operations. This concept, the core of Kolmogorovian probability theory, has proved to be extremely useful, and it plays an important role both in theoretical and in the light of quantum theory. Classical measure theory however needed modification and extension so as to be made suitable as a framework for quantum probability. As a consequence, noncommutative measure theory also called quantum measure theory evolved, which is essentially based on operators rather than scalar measurable functions.