Middle Algebraic K-Theory and L-Theory of von Neumann Algebras

Abstract

So far we have only dealt with the von Neumann algebra N(G) of a group G. We will introduce and study in Section 9.1 the general concept of a von Neumann algebra. We will explain the decomposition of a von Neumann algebra into different types. Any group von Neumann algebra is a finite von Neumann algebra. A lot of the material of the preceding chapters can be extended from group von Neumann algebras to finite von Neumann algebras as explained in Subsection 9.1.4. In Sections 9.2 and 9.3 we will compute K _n (A) and K _n (U) for n = 0,1 in terms of the centers Ƶ(A) and Ƶ(U), where U is the algebra of operators which are affiliated to a finite von Neumann algebra A. The quadratic L-groups L ^∈ _n (A) and L ^∈ _n (U) for n ∈ ℤ and the decorations ∈ = p, h, s are determined in Section 9.4. The symmetric L-groups L ^∈ _n (A) and L ^∈ _n (U) turn out to be isomorphic to their quadratic counterparts.