de Mathématiques de Luminy
Characters of Fredholm modules and a problem of Connes
A large part of research in noncommutative geometry is devoted to various far reaching generalizations of the Atiyah-Singer index theory of elliptic operators on compact manifolds. Following Kasparov [Ka], the notion of an elliptic operator on a manifold is replaced by that of a Fredholm module (or K-cycle) over an algebra of operators on Hilbert space. A number of index theorems for Fredholm modules associated to geometric data have been obtained by Kasparov [Ka], Connes (see [Co2] and the literature cited therein) and others. In order to handle concrete index problems it is not only necessary to dispose of an index theorem but also to provide an index formula which allows the explicit calculation of indices. The classical index formula of Atiyah and Singer is obtained from the index theorem by applying the Chern character in topological K-homology. This motivates the search for character formulas of Fredholm modules (K-cycles) that define a Chern character on the Khomology groups of an algebra of operators. It was this search for a Chern character in K-homology which led A. Connes to the invention of cyclic cohomology [Co1]. He obtained various explicit character formulas [Co1], [Co], [Co2] which depend on the degree of analytic regularity (summability) of the given Fredholm module. Whereas the classical index problems of AtiyahSinger are all finitely summable in the sense of [Co1] there are many examples of Fredholm modules over (noncommutative) algebras which are infinite-dimensional (not finitely summable). Characters of Fredholm modules have been calculated in many finitely summable cases, but as far as the author knows the character of a θ-summable, infinite dimensional (unbounded) Fredholm module [Co] has not yet been determined in a single case. Typical examples of infinite dimensional (unbounded) Fredholm modules are modules over dense subalgebras of the reduced group C∗-algebra C∗r (Γ) of a discrete nonamenable group Γ [Co5], [Co2]. A particularly interesting example is presented by Connes in [Co3], [Co2], where he constructs an infinite dimensional unbounded Fredholm module EΓ over the group ring of a discrete subgroup Γ of a real semisimple Lie group G. This module is closely related to Kasparov’s γ-element [Ka1]. In [Co3], [Co2] Connes makes several steps towards the calculation of the character of EΓ and predicts that it should be cohomologous to the canonical trace on C∗r (Γ). He poses the verification as a problem [Co3],p.83 and notes that a positive solution would imply the Kadison-Kaplansky idempotent conjecture for Γ [Co1], [Co2].