Metrics and spectral triples for Dirichlet and resistance forms

The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet form, we consider Dirac operators within the framework of differential 1-forms proposed by Cipriani and Sauvageot, and comment on its spectral properties. If the Dirichlet form admits a carr\'e operator and the generator has discrete spectrum, then we can construct a related spectral triple, and in the compact and strongly local case the associated Connes distance coincides with the intrinsic metric. We finally give a description of the intrinsic metric in terms of vector fields

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup T on a Banach space X we define its adjoint on an appropriate closed subspace X^o of the norm dual X'. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology (X^o,X). An application is the following: For K a Polish space we consider operator semigroups on the space C(K) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(K) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(K) are precisely those that are adjoints of a bi-continuous semigroups on C(K). We also prove that the class of bi-continuous semigroups on C(K) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if K is not Polish space this is not the case

Extending Whitney's extension theorem: nonlinear function spaces

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where $C$ only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator.Comment: 37 pages, 1 colour figure. v2 small edits, correction to Definition A.3, which makes no impact on proofs or results. Version submitted for publication. v3 small changes in response to referee comments, title extended. v4 crucial gap filled, results not affected. v5 final version to appear in Annales de l'Institut Fourie