Let $N$ be a type $II_{1}$-factor with trace $\tau$.

An $N-N$ correspondence is a Hilbert $N$-bimodule $H$ where the left and right actions are both ultraweakly continuous. Equivalently, a correspondence is given by a *-homomorphism of $N$ into an amplification of $N$. (See Popa's *correspondences* INCREST preprint for more detail.)

The Connes fusion $H \otimes_{N} K$ of correspondences $H$ and $K$ is also called the composition of $H$ and $K$, since it in the "amplification picture" described above, $H \otimes_{N} K$ is isomorphic to the composition of an amplification of the *-homomorphism associated to $K$ with the *-homomorphism associated to $H$.

Furthermore, the Stinespring construction allows us to associate a (cyclic) correspondence $H_{\phi}$ to a completely positive map $\phi$ on $N$, and all cyclic correspondences are of this form, i.e. we can recover the completely positive map from a (unit) cyclic vector. If $\phi_{\xi}$ (resp. $\phi_{\eta}$) denotes the c.p. map associated to a cyclic vector $\xi$ (resp. $\eta$), then it is straightforward to prove that $H_{\phi_{\eta} \circ \phi_{\xi}} \cong H_{\phi_{\xi \otimes_{N} \eta}}$.

It is not true, in general, that $H_{\phi}\otimes_{N} H_{\psi} \cong H_{\psi \circ \phi}$ for normal completely positive maps $\phi$ and $\psi$ on $N$.

Question: Precisely when is $H_{\phi}\otimes_{N} H_{\psi} \cong H_{\psi \circ \phi}$?

It is true if the maps are unital *-homomorphisms, but is false when both maps are $\tau$. Are there any other cases where the statement is true?

Completely bounded maps and operator algebras. Let me know if this did the trick. Thanks for thinking about my problem! – Jon Bannon Mar 24 '11 at 19:18