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This post is a sequel of: Are all the R-R-bimodules completely reducible?

Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely reducible? Is there counter-examples or open cases?

Following the comments of Marcel it seems that $_NM_M$ is completely reducible iff $N' \cap M$ is a type ${\rm I}$ von Neumann algebra. How to prove that in details?

This post is a sequel of: Are all the R-R-bimodules completely reducible?

Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely reducible? Is there counter-examples or open cases?

Following the comments of Marcel it seems that $_NM_M$ is completely reducible iff $N' \cap M$ is a type ${\rm I}$ von Neumann algebra. How to prove that in details?

This post is a sequel of: Are all the R-R-bimodules completely reducible?

Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely reducible? Is there counter-examples or open cases?

This post is a sequel of: Are all the R-R-bimodules completely reducible?

Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely reducible? Is there counter-examples or open cases?

Following the comments of Marcel it seems that $_NM_M$ is completely reducible iff $N' \cap M$ is a type ${\rm I}$ von Neumann algebra. How to prove that in details?

This post is a sequel of: Are all the R-R-bimodules completely reducible?

Let $(N \subset M)$ be a subfactor.

Question: Under which assumptions (as general as possible) on the subfactor, the complete reducibility of the bimodule $_NM_M$ is known to be true, unknown or known to be false?

This post is a sequel of: Are all the R-R-bimodules completely reducible?

Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely reducible? Is there counter-examples or open cases?