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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.

##### MSC:
 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