Sheaf theory and Paschke duality
  Paschke duality identifies the K-homology of a space X with the K-theory of a certain dual C*-algebra. We show that Paschke's dual algebra is in a natural way the algebra of sections of a certain sheaf of C*-algebras over X, which can be thought of as a sheaf of noncommutative symbols. This conceptually simplifies a number of constructions in K-homology, such as the association of a homology class to an elliptic operator and the construction of assembly maps. 
John Roe and Paul Siegel
math.KT