In this paper, we present an analytic definition for the relative torsion for flat C*-algebra bundles over a compact manifold. The advantage of such a relative torsion is that it is defined without any hypotheses on the flat C*-algebra bundle. In the case where the flat C*-algebra bundle is of determinant class, we relate it easily to the L^2 torsion as defined in [7],[5].
We consider a class of nonlocal operators associated with a compact Lie group G acting on a smooth manifold. A notion of symbol of such operators is introduced and an index formula for elliptic elements is obtained. The symbol in this situation is an element of a noncommutative algebra (crossed product by G) and to obtain an index formula, we define the Chern character for this algebra in the framework of noncommutative geometry.
We compute the -theory of -algebras generated by the left regular representation of left Ore semigroups satisfying certain regularity conditions. Our result describes the -theory of these semigroup -algebras in terms of the -theory for the reduced group -algebras of certain groups which are typically easier to handle. Then we apply our result to specific semigroups from algebraic number theory.