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Is the following true?

Let $\mathcal N \subset \mathcal M$ be a subfactor. There is a bijective correspondence between the ultraweakly closed subspaces of $\mathcal M$ that are bimodules over $\mathcal N'\cap \mathcal M$, and the ultraweakly closed subspaces of $\mathcal N$.

If the statement is false, is there a simple way to modify it to make it true? I am particularly interested in the case that $\mathcal M$ is of type $\mathrm I$.

If $\mathcal V \subseteq \mathcal N$ is ultraweakly closed, then $\mathcal V (\mathcal N' \cap \mathcal M)$ is a bimodule over $\mathcal N'\cap \mathcal M$. If the subfactor admits a conditional expectation, then this function is injective. (Edit. Jesse points out that the conditional expectation should be ultrweakly continuous.)

Edit. Steven points out that the statement as written is trivially true by a counting argument. Of course, I'm asking about the function $\mathcal V \mapsto \mathcal V \mathcal (\mathcal N'\cap \mathcal M)$, or something similarly natural. He also notes that irreducible subfactors are a counterexample to the bijectivity of $\mathcal V \mapsto \mathcal V(\mathcal N' \cap\mathcal M )$ in general. This leaves a single concrete question:

Let $\mathcal N \subseteq \mathcal B (\mathcal H)$ be a factor. Is the function $\mathcal V \mapsto \mathcal V \mathcal N'$ a bijection between the ultraweakly closed subspaces of $\mathcal N$ and the ultraweakly closed subspaces of $\mathcal B(\mathcal H)$ that are $\mathcal N'$ bimodules?

2 added 743 characters in body

Is the following true?

Let $\mathcal N \subset \mathcal M$ be a subfactor. There is a bijective correspondence between the ultraweakly closed subspaces of $\mathcal M$ that are bimodules over $\mathcal N'\cap \mathcal M$, and the ultraweakly closed subspaces of $\mathcal N$.

If the statement is false, is there a simple way to modify it to make it true? I am particularly interested in the case that $\mathcal M$ is of type $\mathrm I$.

If $\mathcal V \subseteq \mathcal N$ is ultraweakly closed, then $\mathcal V (\mathcal N' \cap \mathcal M)$ is a bimodule over $\mathcal N'\cap \mathcal M$. If the subfactor admits a conditional expectation, then this function is injective.

Edit. Steven points out that the statement as written is trivially true by a counting argument. Of course, I'm asking about the function $\mathcal V \mapsto \mathcal V \mathcal (\mathcal N'\cap \mathcal M)$, or something similarly natural. He also notes that irreducible subfactors are a counterexample to the bijectivity of $\mathcal V \mapsto \mathcal V(\mathcal N' \cap\mathcal M )$ in general. This leaves a single concrete question:

Let $\mathcal N \subseteq \mathcal B (\mathcal H)$ be a factor. Is the function $\mathcal V \mapsto \mathcal V \mathcal N'$ a bijection between the ultraweakly closed subspaces of $\mathcal N$ and the ultraweakly closed subspaces of $\mathcal B(\mathcal H)$ that are $\mathcal N'$ bimodules?

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# Subspaces of a Subfactor

Is the following true?

Let $\mathcal N \subset \mathcal M$ be a subfactor. There is a bijective correspondence between the ultraweakly closed subspaces of $\mathcal M$ that are bimodules over $\mathcal N'\cap \mathcal M$, and the ultraweakly closed subspaces of $\mathcal N$.

If the statement is false, is there a simple way to modify it to make it true? I am particularly interested in the case that $\mathcal M$ is of type $\mathrm I$.

If $\mathcal V \subseteq \mathcal N$ is ultraweakly closed, then $\mathcal V (\mathcal N' \cap \mathcal M)$ is a bimodule over $\mathcal N'\cap \mathcal M$. If the subfactor admits a conditional expectation, then this function is injective.