A C-correspondence over a C-algebra $A$ is a (right) Hilbert $A$-module (so a Hilbert space) $H$ together with a faithful representation $A\rightarrow B(H)$. Am I right in understanding that a C*-correspondence is just a Hilbert space that you get from the Gelfand-Naimark theorem?
Just to know more, is it possible to have multiple C-correspondences over a single C-algebra? I think yes. I would be happy to see some examples. I am also interested in knowing about any literature where they consider a family of C*-correspondences for whatever purposes.
Let $A,B$ be $C^*$-algebras. An $A$-$B$-$C^*$-correspondence consists of a right Hilbert $B$-module $\mathcal{E}$ together with a (non-degenerate) $*$-homomorphism $$\pi: A \to \mathcal{L}_B(\mathcal{E}).$$ Thus, $A$-$\mathbb{C}$-correspondences correspond to (non-degenerate) $*$-representations of $A$ on Hilbert spaces. This motivates the notion of $C^*$-correspondences, as it is natural to replace Hilbert spaces by Hilbert modules (and sometimes this is really necessary, as the representation theory of $C^*$-algebras on Hilbert spaces is not always strong enough to capture relevant information).
The underlying idea here is simple: $\mathcal{E}$ has (by definition) a right $B$-action. The existence of the $*$-morphism $\pi$ means that $\mathcal{E}$ also carries a left $A$-action, namely $$a.\xi :=\pi(a)\xi, \quad a \in A, \xi \in \mathcal{E}$$ that is compatible with the right $B$-module structure. In other words, an $A$-$B$-$C^*$-correspondence should be thought of some kind of $C^*$-algebraic version of the notion of $A$-$B$-bimodule. Note however the asymmetry that $\mathcal{E}$ does not come with an $A$-valued inner product.