algebra—the elements of the Gelfand spectrum of $A$ [@pedersen:analysisnow]. This leads to our first characterization of when a (nonunital) Frobenius algebra in ${\ensuremath{\mathbf{Hilb}}}$ corresponds to an orthonormal basis.

A Frobenius algebra in ${{\ensuremath{\mathbf{Hilb}}}}$ admits the structure theorem and hence corresponds to an orthonormal basis if and only if it is semisimple.

We first consider sufficiency. Form a direct sum of one-dimensional coalgebras indexed by the copyables of $(A,\delta)$. This will have an isometric embedding as a coalgebra into $(A,\delta)$: $$e \colon \bigoplus_{\{a \mid \delta(a)=a {\ensuremath{\otimes}}a\}} ({\ensuremath{\mathbb{C}}},
    \delta_{{\ensuremath{\mathbb{C}}}}) \to (A,\delta).$$ The image $S$ of $e$ is a closed subspace of $A$, and has an orthonormal basis given by the images of the characters of $A$ qua copyables. The structure theorem holds if the image of $e$ spans $A$.

Given $a \in A$ and a character $c$, the evaluation $c(a)$ gives the Fourier coefficient of $a$ at the basis element of $S$ corresponding to $c$. Now $S$ will be the whole of $A$ if and only if distinct vectors have distinct projections on $S$, [*i.e.* ]{}if and only if distinct vectors have distinct Gelfand transforms $\hat{a} \colon c
  \mapsto c(a)$. Hence the Ambrose structure theorem holds when the Gelfand representation is injective, 