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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).

flag	asked Nov 2 2011 at 11:48 mohanravi 164●4

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welcome, mohan! – Jon Bannon Nov 2 2011 at 11:50

Thanks to Matt for pointing out Takesaki's result. As Jon pointed out, I was wondering if there is a usable criterion that allows us to conclude that a given masa inside a type III factor admits a normal conditional expectation. Incidentally, Cartan masas, by definition, admit normal conditional expectations and there are many type III factors with Cartan masas.(Thanks to Stuart White for this comment). – mohanravi Nov 3 2011 at 0:56

Matthew Daws · Answer 1 · 2011-11-02 20:33:52Z UTC

Takesaki showed in section 6 of:

MR0303307 (46 #2445) Takesaki, Masamichi Conditional expectations in von Neumann algebras. J. Functional Analysis 9 (1972), 306–321. http://www.sciencedirect.com/science/article/pii/0022123672900043

that the following are equivalent for a von Neumann algebra M (not necessarily a factor):

M is finite
Every MASA in M admits a conditional expectation (i.e. norm one normal projection) onto it.

Edit: 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!

Conditional expectations onto masas in type III factors

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