Conditional expectations onto masas in type III factors - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:17:04Z http://mathoverflow.net/feeds/question/79821 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79821/conditional-expectations-onto-masas-in-type-iii-factors Conditional expectations onto masas in type III factors mohanravi 2011-11-02T11:48:02Z 2011-11-02T20:33:52Z <p>There is always a (unique)normal condition expectation onto a masa in a type II_1 factor. When does a masa in a type III factor admit a normal conditional expectation? (If we drop normality, conditional expectations always exist because abelian subalgebras are injective Banach spaces). </p> http://mathoverflow.net/questions/79821/conditional-expectations-onto-masas-in-type-iii-factors/79856#79856 Answer by Matthew Daws for Conditional expectations onto masas in type III factors Matthew Daws 2011-11-02T18:28:26Z 2011-11-02T20:33:52Z <p>Takesaki showed in section 6 of:</p> <p>MR0303307 (46 #2445) Takesaki, Masamichi Conditional expectations in von Neumann algebras. J. Functional Analysis 9 (1972), 306–321. <a href="http://www.sciencedirect.com/science/article/pii/0022123672900043" rel="nofollow">http://www.sciencedirect.com/science/article/pii/0022123672900043</a></p> <p>that the following are equivalent for a von Neumann algebra M (not necessarily a factor):</p> <ul> <li>M is finite</li> <li>Every MASA in M admits a conditional expectation (i.e. norm one normal projection) onto it.</li> </ul> <p><strong>Edit:</strong> As Jon Bannon helpfully points out, the original question asked "when does a MASA admit a conditional expectation onto it", and so this answer only says "not always" which isn't really a full answer!</p>