The generalized Chern character and Lefschetz numbers in \(W^*\)-modules (Q5953562)

Let \(A\) be a von Neumann algebra. In an earlier paper, the author introduced the Grothendieck groups \(N_0(A)\) of equivalence classes of normal elements of \(M_\infty(A) := \lim_{n\to\infty}M_n(A)\). Here two normal elements \(a,b\in A\) are considered as equivalent if their spectral projections \(P_S(a)\), \(P_S(b)\) define the same element in the \(K\)-theory group \(K_0(A)\), for each Borel subset \(S\subset \mathbb{C}\) satisfying \(0\notin \overline{S}\). The groups \(N_0(-)\) define a functor from the category of von Neumann algebras to the category of abelian groups, and there is an obvious natural transformation \(K_0\to N_0\). In the present paper, the author constructs a natural Chern characters homomorphism \(ch : N_0(-)\to H P^{\text{cont}}_0(-)\) to continuous periodic cyclic homology such that the composition \[ K_0(-)\to N_0(-)\to HP^{\text{cont}}_0(-) \] coincides with the usual cyclic Chern character. It should be noted, however, that the cyclic homology of \(C^*\)- and von Neumann algebras is usually rather degenerate and often coincides with the complete commutator quotient of the given algebra. Finally, the author shows that to each unitary endomorphism \(U\) of an elliptic complex \((E, d)\) of \(A\)-bundles over a smooth compact manifold one can associate a Lefschetz number \({\mathcal L}(U, E, d)\) in \(N_0(A)\).