Conditional expectations onto masas in type III factors

Question

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).

Answer 1

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!

Answer 2

It makes sense in the type III setting to ask about a normal conditional expectation onto a subalgebra of a type III factor with respect to a given faithful normal semifinite weight on that factor which has semifinite restriction onto the subalgebra. The existence of such a conditional expectation onto a subalgebra is equivalent to that subalgebra being invariant under the modular automorphism group associated to the weight. (It's Theorem 4.2 on page 211 of Takesaki volume II.)

Although this answers the question in some cheap way, it's probably best viewed as a reprhasing of the question as something like "is there a natural description of the invariance of a MASA in a type III factor under the modular automorphism group of a normal semifinite weight".

I hope this rephrasing is helpful (as you certainly knew about what I said in the first paragraph).