# Uniqueness of the tensor product decomposition of subfactors

A subfactor $(N \subset M)$ is indecomposable if (for $N_i \subset M_i)$: $$(N \subset M) = (N_1 \otimes N_2 \subset M_1 \otimes M_2) \Rightarrow \exists i \ N_i = M_i$$ Then, a subfactor $(N \subset M)$ decomposes into a tensor product of indecomposable subfactors:$$(N \subset M) = (\bigotimes_i N_i \subset \bigotimes_i M_i)$$with $(N_i \subset M_i)$ indecomposable.

Question: Is this decomposition unique for the finite index finite depth irreducible subfactors (up to permutation and isomorphism)?

Remark: The finite group subfactor case comes from the Krull–Schmidt theorem. It generalizes into the Kurosh-Ore theorem in the general theory of modular lattices, with a specific relevant additional result if the lattice is distributive. Perhaps we can use this theorem for answering the question (at least when the lattice of intermediate subfactors $\mathcal{L}(N\subset M)$ is distributive).