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 MASA is equivalent to that MASA 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, 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).

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).