Noncommutative Potential Theory
Noncommutative Potential Theory (NPT) has been developed to provide analytical tools to treat problems arising in Quantum Probability such as the construction of generators of Quantum Dynamical Semigroups and QuantumStochastic Processes. In particular, for example, Dirichlet forms on von Neumann and C*-algebras allowed the constructions of dissipative dynamics to approach equilibria of quantum spin systems (B. Zegarlinski, Y.M. Park). More recently, Dirichlet forms and their underlying differential calculus have been studied to construct Levy processes on Compact Quantum Groups with application to their nuclearity as Banach algebras. Promising researches pertaining NPT will focus on the relations among finite energy states and their potentials on one side and suitable corresponding notions in Quantum Probability on the other side.