$C^{*}$-correspondences viewed as generalized endomorphisms I've heard that $C^{*}$-correspondences (over a $C^{*}$-algebra) can be viewed as generalized endomorphisms of the algebra. I would like to understand this, and be pointed towards books or papers where this is discussed.
Thank you 
 A: Expanding my comment as per suggestion above. All this is explained in more detail in the reference I gave though.
A correspondence from $A$ to $B$ is a Hilbert $B$-module on which $A$ acts non-degenerately by adjointable operators. One can think of this, roughly, as an $A$-$B$-bimodule ${}_{A}\mathcal H_{B}$. These can be composed via tensor product:
$${}_{B}\mathcal K_{C}\circ{}_{A}\mathcal H_{B}={}_{A}(\mathcal H\otimes_B\mathcal K)_{C}.$$
Obviously, this composition will not be associative (or unital), even on the underlying sets. But there are isomorphisms that identify what is supposed to agree. These isomorphisms are not only natural but also satisfy some so-called coherence laws. All this is axiomatized by the notion of a weak $2$-category (or bicategory). If you consider isomorphism classes of correspondences, you get an honest category. But then you will have forgotten about the naturality and coherence of the associator and the unitors.
How does this generalize morphisms of C$^\ast$-algebras and their composition?  A morphism from $A$ to $B$ is a non-degenerate $^\ast$-homomorphism $\varphi$ from $A$ to the multiplier algebra of $\mathcal M(B)$ of $B$, which is also the algebra of adjointable operators on the Hilbert $B$-module $B$. Thus $\varphi$ defines a correspondence $\mathcal H_\varphi$ from $A$ to $B$. Now, if $\varphi$ is a morphism from $A$ to $B$ and $\psi$ is a morphism from $B$ to $C$, then there is a natural isomorphism
$$\mathcal H_\psi\circ\mathcal H_\varphi\cong\mathcal H_{\psi\circ\varphi}.$$
Hence $\varphi\mapsto[\mathcal H_\varphi]$ defines a functor from the category of C$^\ast$-algebras and morphisms to the category of isomorphism classes of correspondences.
