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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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On a smooth manifold tangent bundle can be thought of as a space of its sections, i.e. vector fields. Usually they supposed to be smooth or continuous, but, more generally, one can consider measurable sections (vector fields consisted of vectors, measurably dependent on the point of manifold). One can also equip this space of measurable vector fields with the (pre)inner product: $$ \langle v,u \rangle_{L^2}:=\int_M \langle v_x, u_x \rangle_g d\mathrm{vol}, $$ where $\langle \cdot, \cdot \rangle_g$ is just a Riemannian metric at $x$ coupled with two vectors. Take only vector fields of the finite $||\cdot||_{L^2}$-norm. Completion and separation of the space of these sections make it into a Hilbert space of square-integrable vector fields. This is a prototype (a particular example, to be precise) of the Hilbert space from the paper.

Note, that this Hilbert space can be naturally equipped with a module structure other the algebra $C_0(M)$ of continuous "vanishing at infinity" functions (if M is compact, $C_0(M)=C(M)$). We can just multiply function and vector field fiberwise ("pointwise"), and the resulting vector fields be again square-integrable. Thus, our tangent Hilbert space is a module over $C_0(M)$ and it can also be considered as a bimodule with the same left and right actions.

The tangent bimodule of Cipriani and Sauvageot is a kind of generalization of this structure.

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