reminder: prof. Cipriani's seminar at SISSA

Emanuele Tuillier Illingworth tuillier at
Mon May 5 09:34:05 CEST 2014


Fabio Cipriani
(Politecnico di Milano)

Title: "Non commutative potential theory"

Abstract: In this seminar we illustrate various aspects of the
potential theory of
Dirichlet forms $(E,F)$ on non necessarely commutative C^*-algebras
with trace (A,tau). In particular we introduce finite-energy states,
potentials P and multipliers M(F) of Dirichlet spaces. We will
sketch the proof of the Deny's embedding and Deny's inequality, the
fact that the carrè du champ of bounded potentials are finite-energy
functionals and the fact that, if the trace is finite, bounded
eigenvectors are multipliers. Deny's embedding and the Deny's inequality
are also crucial to prove that the algebra of finite energy multipliers
FcapM(F) is a form core for (E,F) and that it is dense in A
provided the resolvent has the Feller property. We then show the
construction of a spectral triple on the Lipschiz algebra and, more
generally, on the multipliers algebra and why the latter represents a
conformal geometry undelying the Dirichlet space.

Examples include Dirichlet spaces on group C^*-algebras associated to
negative definite functions, Dirichlet spaces on von Neumann algebras
with the Haagerup property, Dirichlet forms arising in Free Probability,
Dirichlet forms on algebras associated to aperiodic tilings, Dirichlet
forms of Markovian semigroups on locally compact spaces, in particular
on post critically finite self-similar fractals, Bochner and Hodge-de
Rham Laplacian on Riemannian manifold.

Venue: Monday 5 May at 4 pm in lecture room 136

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