Amenability and subexponential spectral growth rate of Dirichlet forms   on von Neumann algebras
  In this work we apply Noncommutative Potential Theory to prove (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. 
Fabio Cipriani, Jean-Luc Sauvageot
math.OA