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

where dz is a Lebesgue measure.

0.0.1 Remarks

1. The multiplication “∗” is exactly the composition law that one obtains by considering each point a ∈ C_c(G) as the Schwartz kernel of an operator a on L²(ℝⁿ). Such operators with certain continuity conditions can be realized by kernels that are (Dirac) distributions, or generalized functions on ℝⁿ × ℝⁿ.
2. C_c(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 onto each factor ℝⁿ is a proper map.