INTERDISCIPLINARY STOCHASTIC PROCESSES COLLOQUIUM Tuesday April 20, room 60 Evans, 4.10 - 5.00pm Speaker: Bas Kleijn Title: Complementarity theorems for quantum measures Abstract: The complementarity principle refers to the interpretation of quantum physics by classical observers. In the present context, it gives physical meaning to theorems concerning the relations that exist between quantum and ordinary stochastics. From a mathematical perspective, they allow one to identify the building blocks suitable for a measure-theoretic approach to quantum stochastics analogous to measure-theoretic probability. Guided by complementarity, we define a quantum measure (with appropriately restricted domain, a non-commutative analog of a sigma- algebra), measurability of observables, integrals with respect to a quantum measure, almost-sure equality, non-commutative L_p-spaces etc. Where possible, generalizations of important theorems in measure-theory are given (for instance, the monotone class theorem for measurable selfadjoint operators). To demonstrate the usefulness of complementarity theorems and other formal developments, we conclude with a non-commutative generalization of the Radon-Nikodym theorem (which is also valid in non-separable Hilbert spaces).