Derivations as square roots of Dirichlet forms

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

“…, on the Hilbert space l 2 (Γ), considered as the standard Hilbert space of the left von Neumann algebra L(Γ) generated by the left regular representation of Γ (see [CS1], [C2]). The associated Markovian semigroup is simply given by the multiplication operator [Co2]), which are C * -algebras generated by two unitaries u and v, satisfying the relation vu = e 2iπθ uv , the heat semigroup {T t : t ≥ 0} defined by…”

“…e) There exists a general interplay between Dirichlet forms and differential calculus on tracial C * -algebras (A, τ ) (see [S 2,3], [CS1]) and this provides a source of Dirichlet forms on von Neumann algebras (generated by A in the G.N.S. representation of the trace).…”

Spectral triples for nested fractals

“…The idea of constructing a spectral triple on a fractal as a countable direct sum of finite dimensional spectral triples has been extended from fractals to general compact metric spaces in [33], where the Hausdorff dimension and measure, and the metric were recovered from their noncommutative analogues. More recently,in [20] a different approach to constructing spectral triples on metric spaces was taken, based on [10,11], so the starting point is a regular symmetric Dirichlet form on a locally compact separable metric space, endowed with a nonnegative Radon measure, and the intrinsic (or Carnot-Carathéodory) metric is recovered.…”

“…The idea of constructing a spectral triple on a fractal as a countable direct sum of finite dimensional spectral triples has been extended from fractals to general compact metric spaces in [33], where the Hausdorff dimension and measure, and the metric were recovered from their noncommutative analogues. More recently,in [20] a different approach to constructing spectral triples on metric spaces was taken, based on [10,11], so the starting point is a regular symmetric Dirichlet form on a locally compact separable metric space, endowed with a nonnegative Radon measure, and the intrinsic (or Carnot-Carathéodory) metric is recovered.The basic requirements for a spectral triple T = (A, H, D), where A is a self-adjoint algebra of operators and D is an unbounded self-adjoint operator, both acting on the Hilbert space H, are the boundedness of the commutators [D, a], a ∈ A, and the compactness of the resolvents of D [13]. Based on these hypotheses, one may associate with…”

Dirichlet forms as Banach algebras and applications

“…We study Dirichlet forms from the viewpoint of the theory of commutative Banach algebras; this approach can be compared to the one illustrated in [Cipriani and Sauvageot 2003], where it was shown how the study of regular Dirichlet forms can be regarded as the study of closed derivations on algebras of continuous functions, taking values in Hilbert modules.…”