$C_cG$

Definition 0.1   $ C_c(\mathsf{G})$ is defined as the class (or space) of continuous functions acting on a topological groupoid $ \mathsf{G}$ with compact support, and with values in a field $ F$ . In most applications it will, however, suffice to select $ \mathsf{G}$ as a locally compact (topological) groupoid $ \mathsf{G}_{lc}$ . Multiplication in $ C_c(\mathsf{G})$ is given by the integral formula:

$\displaystyle (a*b)(x,y) = \int_R^n a(x,z)b(z,y)dz ,$

where $ dz$ is a Lebesgue measure.

Remarks

  1. The multiplication “$ *$ ” is exactly the composition law that one obtains by considering each point $ a \in C_c(\mathsf{G})$ as the Schwartz kernel of an operator $ \widetilde{a}$ on $ L^2 (\mathbb{R}^n)$ . Such operators with certain continuity conditions can be realized by kernels that are (Dirac) distributions, or generalized functions on $ \mathbb{R}^n \times \mathbb{R}^n$ .

  2. $ C_c(\mathsf{G})$ can also be more generally defined with values in either a normed space or any algebraic structure. The most often encountered case is that of the space of continuous functions with proper support, that is, the projection of the closure of $ \left\{x,y)\vert a (x,y) \neq 0 \right\}$ onto each factor $ \mathbb{R}^n$ is a proper map.



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