which holds if and only if the algebra is semisimple.

Necessity is easy to see from the form of a direct sum of one-dimensional algebras, as the lattice of ideals is a complete atomic boolean algebra, where the atoms are the generators of the algebras.

We shall restate the previous theorem in terms of axiom (H), so that we have a characterization that lends itself to categories other than ${\ensuremath{\mathbf{Hilb}}}$.

\[prop:semisimple\] A Frobenius algebra in ${{\ensuremath{\mathbf{Hilb}}}}$ satisfies (H) if and only if it is semisimple, and hence admits the structure theorem.

Semisimplicity of proper [H\*]{}-algebras follows from results in [@ambrose:hstar]. Conversely, $\bigoplus_I ({\ensuremath{\mathbb{C}}}, \mu_{{\ensuremath{\mathbb{C}}}})$ is easily seen to satisfy (H); we can define $x^*$ by taking conjugate coefficients in the given basis.

Categorical formulation
-----------------------

We can recast these results into a categorical form. Recall that there is a functor $\ell^2 \colon {\ensuremath{\mathbf{PInj}}} \to
{\ensuremath{\mathbf{Hilb}}}$ on the category of sets and partial injections [@barr:algebraicallycompact; @heunen:thesis]. It sends a set $X$ to the Hilbert space $\ell^2(X) = \{ \varphi \colon X \to
{\ensuremath{\mathbb{C}}} \mid \sum_{x \in X} |\varphi(x)|^2 < \infty \}$, which is the free Hilbert space on $X$ that is equipped with an orthonormal basis, [*i.e.* ]{}an [H\*]{}-algebra, in a 