Hyperexpectations from injective subfactors of a type $II_1$ factor

Question

Let $M$ be a type $\rm{II}_{1}$ factor with trace $\tau$, acting by the GNS representation on $B(L^{2}(M,\tau))$. Let $R\subset M \subset B(L^{2}(M,\tau))$ be a hyperfinite $\rm{II}_{1}$ subfactor of $M$.

Question: Is there a norm one Banach space projection $\Phi: B(L^{2}(M,\tau))\rightarrow R$ such that $\Phi|_M=\mathbb{E}_{R}$, where $\mathbb{E}_{R}$ is the $\tau$-preserving conditional expectation of $M$ onto $R$?

(I actually expect this is never possible, but have been unable to prove this. I figured this should be well-known, so a reference would be a fine answer!)

Answer 1

Such a projection $\Phi$ actually always exists. In the terminology of Definition 2.2 of [N. Ozawa and S. Popa, On a class of II$_1$ factors with at most one Cartan subalgebra. Ann. of Math. 172 (2010), 713-749] the existence of $\Phi$ is saying that $R$ is amenable relative to $\mathbb{C} 1$ inside $M$ (see Theorem 2.1.(3) of the same paper). By Proposition 2.4.(2) of the same paper, this relative amenability holds automatically since $R$ is amenable.

A direct argument can be distilled as follows. Denote by $\hat{x} \in L^2(M)$ the vector that corresponds to $x \in M$. Assume that we have a copy of $M_n(\mathbb{C}) \subset R \subset M$. Denote by $(e_{ij})$ the matrix units of $M_n(\mathbb{C})$. Then $$\Phi_0 : B(L^2(M)) \to M_n(\mathbb{C}) : \Phi_0(T) = \sum_{i,j=1}^n n \, \langle T \hat{e_{j1}},\hat{e_{i1}}\rangle \, e_{ij}$$ defines a conditional expectation of $B(L^2(M))$ onto $M_n(\mathbb{C})$ whose restriction to $M$ equals the unique trace preserving conditional expectation. Generating $R$ by an increasing union of matrix algebras and taking a pointwise weak$^*$ limit point of the corresponding $\Phi_0$ would provide your $\Phi$.