theorem for H\*-algebras, of which we now state the commutative case.

\[thm:ambrose\] Any proper commutative [H\*]{}-algebra (of arbitrary dimension) is isomorphic to a Hilbert space direct sum of one-dimensional algebras.

This is equivalent to asserting isomorphism qua coalgebras. So it is exactly the result we are after! Rather than relying on Ambrose’s results, we now give a direct, conceptual proof, using a few notions from Gelfand duality for commutative Banach algebras.

Copyables and semisimplicity
----------------------------

A *copyable element* of a semigroup $\delta \colon A \to A {\ensuremath{\otimes}}A$ in a monoidal category is a semigroup homomorphism to it from the canonical semigroup on the monoidal unit. More precisely, a copyable element is a morphism $a \colon I \to A$ such that $(a {\ensuremath{\otimes}}a) {\circ}\delta =
\delta {\circ}a$. In a monoidally well-pointed category such as ${\ensuremath{\mathbf{Hilb}}}$, we can speak of a copyable element of $\delta$ as a point $a \in A$ with $\delta(a)=a {\ensuremath{\otimes}}a$.[^8]

\[lem:independent\] Assuming only (A), nonzero copyable elements are linearly independent.

[@hofmann:duality Theorem 10.18(ii)] Suppose that $\{a_0,\ldots,a_n\}$ is a minimal nonempty linearly dependent set of nonzero copyables. Then we can write $a_0$ as $\sum_{i=1}^n
  \alpha_i a_i$ for a suitable choice of coefficients $\alpha_i \in {\ensuremath{\mathbb{C}}}$. So $$\begin{aligned}
        \sum_{i=1}^n \alpha_i 