PlanetPhysics/C cG

$C_c(𝖦)$ is defined as the class (or space) of continuous [[../Bijective/|functions]] acting on a [[../GroupoidHomomorphism2/|topological groupoid]] $𝖦$ with compact support, and with values in a [[../CosmologicalConstant/|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/|formula]]:

$(a*b)(x,y)=\underset{R}{\overset{n}{\int }}a(x,z)b(z,y)dz,$ where $dz$ is a [[../LebesgueMeasure/|Lebesgue measure]].

Remarks

1. The multiplication "$*$" is exactly the [[../Identity2/|composition law]] that one obtains by considering each point

$a\in C_c(𝖦)$ as the Schwartz kernel of an [[../QuantumSpinNetworkFunctor2/|operator]] $\tilde{a}$ on $L^2({ℝ}^n)$. Such [[../QuantumOperatorAlgebra4/|operators]] with certain continuity conditions can be realized by kernels that are (Dirac) distributions, or generalized functions on ${ℝ}^n\times {ℝ}^n$.

1. $C_c(𝖦)$ can also be more generally defined with values in either a normed space or any [[../TrivialGroupoid/|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 $\{x,y)|a(x,y)\ne 0\}$ onto each factor ${ℝ}^n$ is a proper map.