Conditional expectations onto masas in type III factors - MathOverflow most recent 30 from http://mathoverflow.net2013-05-19T00:17:04Zhttp://mathoverflow.net/feeds/question/79821http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/79821/conditional-expectations-onto-masas-in-type-iii-factorsConditional expectations onto masas in type III factorsmohanravi2011-11-02T11:48:02Z2011-11-02T20:33:52Z
<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#79856Answer by Matthew Daws for Conditional expectations onto masas in type III factorsMatthew Daws2011-11-02T18:28:26Z2011-11-02T20:33:52Z<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>