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Geometric \(K\)-theory for Lie groups and foliations. (English) Zbl 0985.46042
This paper is a printed version of a preprint circulated in 1982 with some updated remarks and an extensive updated reference list appended at the end. It outlines with some details a geometrically defined \(K\)-theory involving group (or groupoid) actions, and proposes some important hard (and famous since then) conjectures. As a starting point for noncommutative geometry, the classical topological \(K\)-theory had been canonically adapted and generalized to an analytical \(K\)-theory of C*-algebras (or Banach algebras) which provides a natural context for the indices of elliptic operators, an important ingredient in the Atiyah-Singer index theorem. Although the analytical \(K\)-groups are fairly easy to define, they are often difficult to compute. On the other hand, the geometric \(K\)-groups, e.g. \(K^{\ast }\left( X,G\right) \) for a \(G\)-manifold \(X\), introduced in this paper involve topological machineries, e.g. the Thom isomorphism theorem and Gysin map, but can be more easily computed in some concrete cases than the corresponding analytical \(K\)-groups, e.g. \(K_{\ast }\left( C_{0}\left( X\right) \rtimes _{r}G\right) \) of the reduced cross-product C*-algebra \( C_{0}\left( X\right) \rtimes _{r}G\). A major conjecture proposed is the isomorphism between \(K^{\ast }\left( X,G\right) \) and \(K_{\ast }\left( C_{0}\left( X\right) \rtimes _{r}G\right) \) which is known to be true for proper \(G\)-manifolds \(X\) and is essentially the Atiyah-Singer index theorem in the case of compact groups \(G\). Variations of the geometric \(K\)-group \( K^{\ast }\left( X,G\right) \) involving discrete group \(G\) with a 2-cocycle or holonomy groupoid \(G\) of a foliation are also presented and discussed.

46L87 Noncommutative differential geometry
46L80 \(K\)-theory and operator algebras (including cyclic theory)
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
58B34 Noncommutative geometry (à la Connes)
58J20 Index theory and related fixed-point theorems on manifolds
19K56 Index theory
46L55 Noncommutative dynamical systems