PlanetPhysics/C cG

$$C_c ( \mathsf{G})$$ 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:

$$(a*b)(x,y) = \int_R^n a(x,z)b(z,y)dz ,$$ where $$dz$$ is a Lebesgue measure.

Remarks
$$a \in C_c(\mathsf{G})$$ 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 $$\mathbb{R}^n \times \mathbb{R}^n$$.
 * 1) The multiplication "$$*$$" is exactly the composition law that one obtains by considering each point


 * 1) $$C_c(\mathsf{G})$$ 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 $$\left\{x,y)|a (x,y) \neq 0 \right\}$$ onto each factor $$\mathbb{R}^n$$ is a proper map.