Rocco Duvenhage

Joinings of W*-dynamical systems

Abstract We study the notion of joinings of W ∗ -dynamical systems, building on ideas from measure theoretic ergodic theory. In particular we prove sufficient and necessary conditions for ergodicity in terms of joinings, and also briefly look at conditional expectation operators associated with joinings.

Relatively independent joinings and subsystems of W*-dynamical systems

Relatively independent joinings of W*-dynamical systems are constructed. This is intimately related to subsystems of W*-dynamical systems, and therefore we also study general properties of subsystems, in particular fixed point subsystems and compact subsystems. This allows us to obtain characterizations of weak mixing and relative ergodicity, as well as of certain compact subsystems, in terms of joinings.

The Nature of Information in Quantum Mechanics

A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observers information regarding a physical system. This is seen as the main difference from classical mechanics, where an observers information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process.

Noncommutative Ricci flow in a matrix geometry

We study noncommutative Ricci flow in a finite-dimensional representation of a noncommutative torus. It is shown that the flow exists and converges to the flat metric. We also consider the evolution of entropy and a definition of scalar curvature in terms of the Ricci flow.

Recurrence in Quantum Mechanics

We first compare the mathematical structure of quantum and classical mechanics when both are formulated in a C*-algebraic framework. By using finite von Neumann algebras, a quantum mechanical analogue of Liouvilles theorem is then proposed. We proceed to study Poincaré recurrence in C*-algebras by mimicking the measure theoretic setting. The results are interpreted as recurrence in quantum mechanics, similar to Poincaré recurrence in classical mechanics.

Bergelson's theorem for weakly mixing C*-dynamical systems

We study a nonconventional ergodic average for asymptotically abelian weakly mixing C*-dynamical systems, related to a second iteration of Khintchines recurrence theorem obtained by Bergelson in the measure theoretic case. A noncommutative recurrence theorem for such systems is obtained as a corollary.

Recurrence and ergodicity in unital ∗-algebras

Abstract Results concerning recurrence and ergodicity are proved in an abstract Hilbert space setting based on the proof of Khintchines recurrence theorem for sets, and on the Hilbert space characterization of ergodicity. These results are carried over to a non-commutative ∗ -algebraic setting using the GNS-construction. This generalizes the corresponding measure theoretic results, in particular a variation of Khintchines theorem for ergodic systems, where the image of one set overlaps with another set, instead of with itself.

Noncommutative recurrence over locally compact Hausdorff groups

We extend previous results on noncommutative recurrence in unital *-algebras over the integers to the case where one works over locally compact Hausdorff groups. We derive a generalization of Khintchines recurrence theorem, as well as a form of multiple recurrence. This is done using the mean ergodic theorem in Hilbert space, via the GNS construction.

A mean ergodic theorem for actions of amenable quantum groups

We prove a weak form of the mean ergodic theorem for actions of amenable locally compact quantum groups in the von Neumann algebra setting. 2000 MSC: Primary 46L55; Secondary 37A30

Detailed balance and entanglement

We study a connection between quantum detailed balance, which is a concept of importance in statistical mechanics, and entanglement. We also explore how this connection fits into thermofield dynamics.