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.
KeywordsCartan Subalgebra Holonomy Group Unit Space Chern Character Transverse Measure
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