On the K- and L-theory of the algebra of operators affiliated to a finite von Neumann algebra
type: Item
instance of: scholarly article
title: On the K- and L-theory of the algebra of operators affiliated to a finite von Neumann algebra
zbMATH Open document ID: 0998.19005
DOI: 10.1023/A:1014078228859
author: Holger Reich
published in: \(K\)-Theory
publication date: 2002-08-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 19K99
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 19G24
zbMATH DE Number: 1785680
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: \(L\)-theory
zbMATH Keywords: von Neumann algebras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1785680