  Heat kernels are used in this paper to express the analytic index of
projectively invariant Dirac type operators on G-covering spaces of compact
manifolds, as elements in the K-theory of certain unconditional completions of
the twisted group algebra of G. This is combined with V. Lafforgue's results in
the untwisted case, to compute the range of the trace on the K-theory of these
algebras, under the hypothesis that G is in the class C' (defined by V.
Lafforgue).
