Does there exist a classification, or characterization, of finite-dimensional Hilbert $C^\*$-modules? More generally, does there exist a characterization of countable direct sums of finite-dimensional Hilbert $C^\*$-modules


Edit: By finite dimensional I mean that the $C^\*$-module is finite dimensional as a vector space.



Finite-dimensional Hilbert $C^*$-modules
As a vector space, okay. Let $A$ be a C\*-algebra and $E$ a Hilbert module over $A$ which is finite dimensional as a vector space. Then $A$ acts by left multiplication on $E$, i.e., we have a bounded homomorphism from $A$ into $B(E) \cong M\_n$. Letting $I$ be the kernel of this homomorphism, this shows that $A$ has a finite codimension closed ideal $I$ (which is automatically self-adjoint) and $E$ is a Hilbert module over $A/I$ (exercise).


So we have to classify finite dimensional Hilbert modules over finite dimensional C\*-algebras. Any finite dimensional C\*-algebra is a direct sum of matrix algebras, and any Hilbert module over such a direct sum decomposes as a direct sum of Hilbert modules over the summand matrix algebras. So we only have to classify the finite dimensional Hilbert modules over $M\_n$. But again, left multiplication gives a homomorphism from $M\_n$ into $B(E)$, and this $\*$-homomorphism must be unital (exercise), which by the answer to [this question](https://mathoverflow.net/questions/32511/homomorphisms-between-matrix-algebras) means that $E \cong \mathbb{C}^n\otimes \mathbb{C}^k$ for some $k$, with the action being $a(v\otimes w) = av\otimes w$. So $E$ is a direct sum of finitely many copies of $\mathbb{C}^n$ as a module over $M\_n$ in the natural way. The inner product is described in Matthew's answer.


