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(Definition)

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

is a Lebesgue measure.

Remarks

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 $\mathbb{R}^n \times \mathbb{R}^n$ .
$C_c(\mathsf{G})$ 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.