Amenability and subexponential spectral growth rate of Dirichlet forms on von Neumann algebras

In this work we apply Noncommutative Potential Theory to characterize (relative) amenability and the (relative) Haagerup Property (H) of von Neumann algebras in terms of the spectral growth of Dirichlet forms. Examples deal with (inclusions of) countable discrete groups and free orthogonal compact quantum groups.