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A paper by Cipriani and Sauvageot, available at

http://dx.doi.org/10.1016/S0022-1236(03)00085-5

shows that for many Dirichlet forms on $C^*$-algebras there is a derivation $\delta$ from the domain of the form to a Hilbert module $H$ so that $\|\partial a\|^2_H = E(a,a)$ (for $E$ the Dirichlet/energy form).

They repeatedly refer to $H$ as the "tangent module" or "tangent bimodule," and it is this usage of tangent that has me curious. But this paper is frankly above my level and I am struggling to make sense of it all. It brings to my mind connections with noncommutative geometry, and the authors note in a few cases that, for instance,

  • "As far as the commutative case of algebras of continuous functions is concerned, our algebraic approach to the differential calculus on measured metric spaces apply to the Dirichlet forms constructed by Sturm and is a version of the one constructed more analytically by Cheeger. In this respect, assuming a doubling property and a weak Poncare inequality for the measure, he is able to prove that the measurable tangent bundle is finite dimensional. In algebraic terms, the bimodule $H$ we construct has finite multiplicity" (Pg. 79).

Can anybody clarify the relation between tangent bundles and this Hilbert module in a very down-to-earth way?

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