Algebraic \(K\)-theory of von Neumann algebras (Q1319259)

The paper is devoted to the computation of the algebraic \(K\)-group \(K_ 1({\mathcal A})\) and a closely related group \(K_ 1^ w ({\mathcal A})\) of a von Neumann algebra \({\mathcal A}\). First the authors define the algebraic \(K\)-group \(K_ 1 ({\mathcal A})\) as usual as being generated by bijective endomorphisms of finitely generated projective \({\mathcal A}\)-modules with certain relations, and the group \(K_ 1^ w ({\mathcal A})\) as being generated by injective endomorphisms of such modules and the same relations. Next, it is shown that it suffices to compute these groups in the case where \({\mathcal A}\) is of some fixed type \((\text{I}_ f\), \(\text{I}_ \infty\), \(\text{II}_ 1\), \(\text{II}_ \infty\) or III). In the type \(\text{I}_ f\) case, where \({\mathcal A}\) is a product of matrix algebras over abelian von Neumann algebras, the authors show that a certain normalization of the usual determinant induces a bijection of \(K_ 1 ({\mathcal A})\) with the invertible elements in the center of \({\mathcal A}\) and of \(K_ 1^ w ({\mathcal A})\) with the Grothendieck group of the semigroup of all elements \(a\) in a center of \({\mathcal A}\) such that multiplication with \(a\) is an injective mapping. Next, it is shown that for countably composable von Neumann algebras of type \(\text{II}_ 1\) the so called Fuglede-Kadison determinant defines an isomorphism of \(K_ 1 ({\mathcal A})\) with the positive invertible elements in the center of \({\mathcal A}\), and that in this case the group \(K_ 1^ w({\mathcal A})\) is trivial. Then the authors prove that in all other cases, i.e. for properly infinite von Neumann algebras both groups are trivial. Finally, it is shown how the results obtained so far can be used to detect nontrivial elements in the Whitehead groups of finite groups.