On the K- and L-theory of the algebra of operators affiliated to a finite von Neumann algebra
Let \(\mathcal A\) be a finite von Neumann algebra and let \(\mathcal U\) be the ring of all closed densely defined operators affiliated to \(\mathcal A\). Then \(\mathcal U\) has pleasant ring-theoretic properties. It is a von Neumann regular ring and it is the Ore localization of \(\mathcal A\) with respect to its non zerodivisors. Thus the category of \(M\)-modules can be identified as the quotient of the category of \(\mathcal A\)-modules modulo the natural category \(\mathcal T\) of torsion modules. It follows that \(K_0({\mathcal A})\cong K_0({\mathcal U})\), and a dimension function gives an embedding \(K_0({\mathcal U}) \to \mathbb{R}\). Computations of \(K_1({\mathcal A})\) and \(K_1({\mathcal U})\) are given, and \(K_0({\mathcal T})\) is recovered from the localization sequence. The corresponding results for \(L\)-groups are also discussed.
scientific article; zbMATH DE number 1785680