interest is, of course, in the nonunital case. To explain the provenance of the (H) axiom, and its implications for obtaining a correspondence with orthonormal bases in ${\ensuremath{\mathbf{Hilb}}}$ in arbitrary dimension, we shall now study the situation in the concrete setting of Hilbert spaces.

[H\*]{}-algebras in ${\ensuremath{\mathbf{Hilb}}}$ {#sec:hstarinhilb}
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We begin by revisiting Theorem \[cpvth\]. How should the correspondence between Frobenius algebras and orthonormal bases be expressed mathematically? In fact, the content of this result is really a *structure theorem* of a classic genre in algebra [@albert:structureofalgebras]. The following theorem, the *Wedderburn structure theorem*, is the prime example; it was subsequently generalized by Artin, and there have been many subsequent developments.

Every finite-dimensional semisimple algebra is isomorphic to a product of full matrix algebras. In the commutative case over the complex numbers, this has the form: the algebra is isomorphic to a product of one-dimensional complex algebras.

To see the connection between the Wedderburn structure theorem and Theorem \[cpvth\], consider the coalgebra $A$ determined by an orthonormal basis $\{ {{|} i\rangle} \}$ on a Hilbert space: $$\delta \colon {{|} i\rangle} \mapsto {{|} ii\rangle}.$$ This is isomorphic as a coalgebra to a direct sum of one-dimensional coalgebras $$\delta_{{\ensuremath{\mathbb{C}}}} \colon {\ensuremath{\mathbb{C}}} \to 