Quantum dynamics, both reversible (i.e., closed quantum systems) and
irreversible (i.e., open quantum systems), gives rise to product
Hilbert spaces or, more generally, of Hilbert modules. When we consider
reversible dynamics that dilates an irreversible dynamics, then
system of the latter is equal to the product system of the former (or
contained in a unique way). Whenever the dynamics is on a proper
subalgebra of the algebra of all bounded
operators on a Hilbert space, in particular, when the open system is
classical (commutative) it
is indispensable that we use Hilbert modules.
The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally independent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or module determines the sort of quantum independence underlying the reversible system.
This school brings together experts from quantum dynamics, product systems and quantum independence and young researchers who want to learn more about this field.
Institut für Mathematik und Informatik
Division for Algebra and Functional Analytic Applications