# Is a C*-correspondence just a Hilbert space you get from the Gelfand-Naimark theorem?

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.

• A GNS representation doesn’t have to be faithful. And a single representation can be a direct sum of multiple GNS representations. But it is certainly true that every nondegenerate representation of a $C^\ast$-algebra is a direct sum of GNS representations. Feb 2 at 22:37
• A $C^\ast$-algebra can, of course, have multiple non-equivalent faithful representations. Say, a faithful cyclic representation cannot be equivalent to a faithful non-cyclic one. Feb 2 at 22:40
• If $A$ is a $C^*$-algebra of dimension $\ge2$, a Hilbert $A$ module is not going to have a (compatible) structure of a Hilbert space. Feb 3 at 3:45
• A Hilbert $A$-module is not a Hilbert space per se. You should check your definitions again! Feb 4 at 10:09
• Awygan and Just dropped in raised a good point. I was following the description in your question statement and assumed we are talking about a Hilbert space, but a Hilbert $A$-modules is not a Hilbert space. While it does have an inner product, the inner product does not take values in $\mathbb{C}$ but instead in $A$. Which definition do you actually want to use? Feb 4 at 21:03

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.