Let $R$ be the hyperfinite $II_1$ factor and let $X$ be any $R$-$R$-bimodule.

Question  : Is $X$ completely reducible (i.e. a direct integral of irreducible $R$-$R$-bimodules)  ?

Example  : Let $(N \subset M)$ an irreducible, finite depth and finite index (hyperfinite $II_1$) subfactor,
then $_NL^{2}(M)_N$ is completely reducible.

Let $R$ be the hyperfinite $II_1$ factor and let $X$ be any $R$-$R$-bimodule.

Question: Is $X$ completely reducible (i.e. a direct integral of irreducible $R$-$R$-bimodules)?

Example: If $(N \subset M)$ is an irreducible, finite depth and finite index (hyperfinite $II_1$) subfactor, then $_NL^{2}(M)_N$ is completely reducible.

Let $R$ be the hyperfinite $II_1$ factor and let $X$ be any $R-R$ bimodule.

Question : Is $X$ completely reducible (i.e. a direct integral of irreducible $R-R$ bimodules) ?

Example : Let $N \subset M$ an irreducible, finite depth and finite index (hyperfinite $II_1$) subfactor,
then $_NL^{2}(M)_N$ is completely reducible.

Let $R$ be the hyperfinite $II_1$ factor and let $X$ be any $R$-$R$-bimodule.

Question : Is $X$ completely reducible (i.e. a direct integral of irreducible $R$-$R$-bimodules) ?

Example : Let $(N \subset M)$ an irreducible, finite depth and finite index (hyperfinite $II_1$) subfactor,
then $_NL^{2}(M)_N$ is completely reducible.