More on bimodules

Suppose M, P are arbitrary von Neumann algebras with separable pre-duals, and suppose is a separable M-P-bimodule. Pick some faithful normal state φ and set and. It follows from Theorem 2.2.2 that may be identified, as a left M-module, with for some projection q ∈ M∞(M) (which is uniquely determined up to Murray–von Neumann equivalence in M∞ (M)); further, we have. Since is an M-P-bimodule, it follows from our identification that there exists a normal unital homomorphism θ : P → M∞(M)q such that the right action of P is given by ξ · y = ξθ(y).

Conversely, given a normal homomorphism θ : P → M∞(M), let denote the M-P-bimodule with underlying Hilbert space, and with the actions given, via matrix multiplication, by m · ξ · p = mξθ(p). The content of the preceding paragraph is that every separable M-P-bimodule is isomorphic to for suitable θ.

If M is a factor of type III, then so is M∞(M), and hence every non-zero projection in M∞x(M) is Murray–von Neumann equivalent to 1M ⊗ e11. Consequently, every M-M-bimodule is isomorphic to rθ for some endomorphism θ : M → M.

Suppose M and P are II1 factors and suppose is as above.