I don't know how general this should be: let $F$ be a finitely generated projective $A-A$ bimodule and $S$ be a finitely generated projective $B-A$ bimodule. Then consider the map $\alpha:T \mapsto id_S \otimes T$ where $T \in End_A(F)$ so $\alpha(T) \in End_B(E)$ where $E:=S \otimes_A F$ where the left $B$ module structure on $E$ is via $b(s \otimes f)=bs \otimes f$. The claim is

The map $\alpha$ is an isomorphism.

I have a problem in proving that $\alpha$ is epimorphism: my context is not the most general, for example $A$ and $B$ are Morita equivalent and $S$ establishes this Morita equivalence. Moreover modules $S,F$ come from some vector bundles therefore they are equipped with the hermitian pairing. I tried to prove that every operator of the form $|s \otimes f \rangle \langle s' \otimes f'|$ lies in the image of $\alpha$ where $|s \otimes f \rangle \langle s' \otimes f'| (s'' \otimes f'')=\langle s'' \otimes f'',s' \otimes f' \rangle (s \otimes f)$ however I don;t see why it is true (note that $(id_S \otimes T)(s'' \otimes f'')$ always will be of the form $s'' \otimes (...)$ which is not the case for our "rank one" operator $|s \otimes f \rangle \langle s' \otimes f'|$).


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