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
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3$\begingroup$ Morphisms induce correspondences and composition extends to correspondences (up to isomorphism). Hence correspondences generalize morphisms. However, composition of correspondences is only associative up to isomorphism. Hence it is natural to view correspondences as a weak 2-category. See arxiv.org/abs/0908.0455 $\endgroup$– RasmusCommented Dec 8, 2015 at 9:10
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1$\begingroup$ @RasmusBentmann: it might be nice to expand this into an answer ... $\endgroup$– Nik WeaverCommented Dec 8, 2015 at 14:50
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$\begingroup$ @RasmusBentmann: Can you please explain how morphisms induce correspondences and how composition extends to correspondences, or point me to a book/paper discussing this ? Thank you $\endgroup$– epsilonCommented Dec 8, 2015 at 17:42
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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.
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$\begingroup$ Thank you for the reply Rasmus. So if I understand correctly, a correspondence over $A$ generalizes a $C^{*}$-endomorphism $\varphi:A \to A$ because now the range of $\varphi$ does not have to be $A$ but it can be a $C^{*}$-algebra containing $A$ (the multiplier algebra of $A$). Does this make sense? $\endgroup$– epsilonCommented Dec 8, 2015 at 23:09
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$\begingroup$ No. First of all, when you say endomorphism, do you mean a $^\ast$-homomorphism from $A$ to $A$ or a morphism from $A$ to $A$? A degenerate $^\ast$-homomorphism will not induce a correspondence. Instead we can consider morphisms from $A$ to $A$ as defined above. This is not unnatural: for commutative C$^\ast$-algebras, these correspond to continuous maps (while $^\ast$-homomorphisms correspond to something else). This has nothing to do with correspondences yet. However, every morphism induces a correspondence. Therefore, we can think of correspondences as a generalization of morphisms. $\endgroup$– RasmusCommented Dec 9, 2015 at 8:22