In many contexts it is helpful to think of conditional expectations as averages of unitary conjugates, a standpoint vindicated by many standard techniques in the theory of finite von Neumann algebras. Further, this point of view is reinforced nicely in a pointwise sense (cf. Narutaka Ozawa's comments to my earlier question here that point to a nice set of references.) The present question should be seen as an attempt to test to what extent this point of view can trusted. In fact, although the present question asks for the substantially weaker requirement of approximate innerness, its intent is to unveil to what extent the view of expectation as average of inner automorphisms can be carried out in a uniform way, i.e. at the level of completely positive maps. This is natural to consider, and a positive answer would provide a basic tool which would be very useful to have, and a counterexample would be of basic enough interest that one should be easier to find than it presently is on the web.

Let $M$ be a type $\textbf{II}_{1}$ factor with trace $\tau$ and $1\in N\subseteq M$ a von Neumann subalgebra.

Question: Is it always the case that the unique $\tau$-preserving conditional expectation $\mathbb{E}_{N}$ of $M$ onto $N$ is approximately inner as a completely positive map on $M$? I.e. is $\mathbb{E}_{N}$ in the point-ultraweak closure of the set of all finite sums of completely positive maps $\varphi: M\rightarrow M$ of the form $\varphi(x)=axa^{*}$ with $a\in M$?

The basic nature of this question suggests that it has been extensively studied. I would heartily welcome a list of references to papers that may be helpful. We note that by results of Anantharaman-Delaroche (pardon the JLMS paywall) the answer to this question is affirmative when M is the hyperfinite $\textbf{II}_{1}$ factor, but for the strong reason that inner completely positive maps from $M$ into $M$ are dense in the cone of all such maps. The present question doesn't seem to require anything nearly so strong, and is closer in spirit to the above philosophy that conditional expectations are unitary averages.

  • $\begingroup$ What about conditional expectations to fixpoint algebras of automorphisms that are not approximately inner? (Like the flip on the tensor product of a non-hyperfinite type $II_1$-factor with itself.) $\endgroup$ – Ulrich Pennig Jul 14 '17 at 20:58
  • $\begingroup$ @Ulrich Pennig: Interesting suggestion. Thanks! I will think about this. $\endgroup$ – Jon Bannon Jul 14 '17 at 23:04

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