Maximal localizations of von Neumann algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:46:59Z http://mathoverflow.net/feeds/question/1940 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1940/maximal-localizations-of-von-neumann-algebras Maximal localizations of von Neumann algebras Dmitri Pavlov 2009-10-22T20:30:14Z 2010-08-05T14:47:27Z <p>Suppose M is a von Neumann algebra. Denote by L its maximal noncommutative localization, i.e., the Ore localization with respect to the set of all left and right regular elements, i.e., elements whose left and right support equals 1.</p> <p>Denote by A the set of all closed unbounded operators with dense domain affiliated with the standard representation of M on a Hilbert space, i.e., L^2(M), also known as the standard form of M.</p> <p>Von Neumann proved that if M is finite, then L and A are canonically isomorphic.</p> <p>What can we say about the relationship of L and A when M has type III?</p> <p>I am also interested in the properly infinite semifinite case.</p> http://mathoverflow.net/questions/1940/maximal-localizations-of-von-neumann-algebras/34635#34635 Answer by Andreas Thom for Maximal localizations of von Neumann algebras Andreas Thom 2010-08-05T14:47:27Z 2010-08-05T14:47:27Z <p>I think the question is not well-posed or has a negative answer.</p> <p>One first has to deal with the question whether the left-right-regular elements satisfy the Ore condition, or equivalently, we have to ask: Can we find common denominators? If one is not in the finite case, this is not possible. </p> <p>For \$B(H)\$, let us take injective bounded operators \$T\$ and \$S\$ such that the images are dense but intersect only in the zero vector. In order to find a common denominator, we need to find an operator \$R\$ (bounded, injective, dense image) and bounded operators \$X\$ and \$Y\$ such that \$R = TX\$ and \$R= SY\$. This cannot possibly work since \$T\$ and \$S\$ have disjoint image.</p> <p>Since \$B(H)\$ sits inside any type \$III\$-factor, no Ore localization in the above sense exists.</p>