  We define N-theory being some analogue of K-theory on the category of von
Neumann algebras such that $K_0(A)\subset N_0(A)$ for any von Neumann algebra
A. Moreover, it turns out to be possible to construct the extension of the
Chern character to some homomorphism from $N_0(A)$ to even Banach cyclic
homology of A. Also, we define generalized Lefschetz numbers for an arbitrary
unitary endomorphism U of an A-elliptic complex. We study them in the situation
when U is an element of a representation of some compact Lie group.
