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

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 $\mathbb {R} ^{n}\times \mathbb {R} ^{n}$.

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.