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

where is a Lebesgue measure.

Remarks

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  1. The multiplication " " is exactly the composition law that one obtains by considering each point

  as the Schwartz kernel of an operator   on  . Such operators with certain continuity conditions can be realized by kernels that are (Dirac) distributions, or generalized functions on  .

  1.   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   is a proper map.