Let $(N \subset M)$ be a finite index irreducible subfactor, and $P = P(N \subset M)$ its planar algebra.

We can see $M$ as an algebraic $N$-$N$-bimodule, it decomposes into irreducible algebraic $N$-$N$-bimodules: $$M = \bigoplus_{i \in I} V_i \otimes B_i$$ with $(B_i)_{i \in I}$ the irreducible algebraic sub-$N$-$N$-bimodules of $M$, and $V_i$ the multiplicity space.

Now $P_{2,+} = N' \cap M_1$ is a finite dimensional ${\rm C}^*$-algebra and by the double characterization of the principal graph, we know that:

$$ P_{2,+} \simeq \bigoplus_{i \in I} End(V_i)$$

The one-to-one correspondence between the projections of $p \in P_{2,+}$ and the algebraic sub-$N$-$N$-bimodules $X_p$ of $M$, comes from the one-to-one correspondence between the projections of $P_{2,+}$ and the subspaces of $V=\oplus_{i \in I}V_i$ on one hand (through image and range projection), and the one-to-one correspondence between the subspaces of $V$ and the algebraic sub-$N$-$N$-bimodules of $M$ on the other hand (by definition of the decomposition of $M$). Moreover this correspondence is an isomorphism of poset, which means $$p \le q \Rightarrow X_p \subseteq X_q$$ The set of biprojections is a subposet of the set of projections and the set of intermediate subfactors is also a subposet of the set of algebraic sub-$N$-$N$-bimodules of $M$ and using this paper of D. Bisch, $$p \text{ is a biprojection iff } X_p \text{ is an intermediate subfactor. } $$

*Corollary*: The biprojection $\langle p \rangle$ generated by the projection $p \in P_{2,+}$ corresponds to the intermediate subfactor $\langle X_p \rangle$ generated by the algebraic sub-$N$-$N$-bimodules $X_p$ of $M$, which can be reformulated by:

$$X_{\langle p \rangle} = \langle X_{ p } \rangle$$
*proof*: First $\langle p \rangle$ is the minimal biprojection $b \ge p$, and next $\langle X_p \rangle$ is the minimal von Neumann algebra containing $X_p$ in $M$, but $N \subset \langle X_p \rangle$ because $X_p$ is a $N$-$N$-bimodule, and so $\langle X_p \rangle$ is an intermediate subfactor by irreducibility of $(N \subset M)$. The result follows by the poset isomorphism respecting the one-to-one correspondence between biprojections and intermediate subfactors $\square$

**Questions**:

- Is there an explicit formula for $X_p$ using $M$, $p$ (and $e^M_N$)?
- Is it true that $X_p^* = X_{\overline{p}}$ (with $x^*$ the dual of $x \in M$ and $\overline{p}$ the contragredient of $p \in P_{2,+}$)?
- Is it true that $Vect_{\mathbb{C}}(X_p X_q) = X_{S(p * q)}$ (with $X_p X_q$ the product of $X_p$ and $X_q$ in $M$, $p*q$ the coproduct of $p$ and $q$, and $S(u)$ the support projection of $u \in P_{2,+}$)?

Marcel has pointed out in comment that for $p$ a biprojection $X_p = \{p \}' \cap M$ (theorem 3.2. p213 of this paper).

Note that if $p$ is not a biprojection then $X_p$ has not an algebra (see the *algebra objects* of a fusion category).