We turn now to the discussion of operator algebras that can be associated with groupoids and in particular to the groupoid of a foliated space. For this discussion we start with a locally compact second countable topological groupoid G and we assume given a continuous tangential measure λ (see Chapter IV for the definition). Thus for each x in the unit space X of G we have a measure λx on Gx = r−1(x) with certain invariance and continuity properties as described in Chapter IV. For the moment we do not need to assume that the groupoid has discrete holonomy groups as in Chapter IV, but all the examples and all the applications will satisfy this condition. If in addition the support of the measure λx is equal to r−1(x), as is usual in our examples, then λ is called a Haar system.

In this chapter we construct the C*-algebra of the groupoid and we determine this algebra in several important special cases. (As general references for C*-algebras, we recommend [Davidson 1996; Fillmore 1996; Arveson 1976; Pedersen 1979].) We describe the Hilsum–Skandalis stability theorem. Assuming a transverse measure, we construct the associated von Neumann algebra and develop its basic properties and important subalgebras. This leads us to the construction of the weight associated to the transverse measure; it is a trace if and only if the transverse measure is invariant.