{\ensuremath{\mathbb{C}}} {\ensuremath{\otimes}}{\ensuremath{\mathbb{C}}}, 
  \qquad 1 \mapsto 1 {\ensuremath{\otimes}}1.$$ To say that a Frobenius algebra corresponds to an orthonormal basis is exactly to say that it is isomorphic as a coalgebra to a Hilbert space direct sum of one-dimensional coalgebras: $$A \; \cong \; \bigoplus_I \, ({\ensuremath{\mathbb{C}}}, \delta_{{\ensuremath{\mathbb{C}}}}),$$ where the cardinality of $I$ is the dimension of $H$. Applying dagger, this is equivalent to $A$ being isomorphic as an *algebra* to the direct sum of one-dimensional *algebras* $$A \; \cong \; \bigoplus_I \, ({\ensuremath{\mathbb{C}}}, \mu_{{\ensuremath{\mathbb{C}}}}), 
  \qquad \mu_{{\ensuremath{\mathbb{C}}}} \colon 1 {\ensuremath{\otimes}}1 \mapsto 1.$$ In this case, we say that the Frobenius algebra *admits the structure theorem*, making the view of bases as (co)algebras precise.

[H\*]{}-algebras {#h-algebras}
----------------

There is a remarkable generalization of the Wedderburn structure theorem to an infinite-dimensional setting, in a classic paper from 1945 by Warren Ambrose on ‘[H\*]{}-algebras’ [@ambrose:hstar]. He defines an [H\*]{}-algebra[^5] as a (not necessarily unital) Banach algebra based on a Hilbert space $H$, such that for each $x \in H$ there is an $x^* \in H$ with $${\ensuremath{\langle xy\,|\,z \rangle}} = {\ensuremath{\langle y\,|\,x^*z \rangle}}$$ for all $y,z \in H$, and similarly for right multiplication. Note that $${\ensuremath{\langle xy\,|\,z \rangle}} 
  = (\mu {\circ}(x {\ensuremath{\otimes}}y))^\dag {\circ}z 
  = (x^\dag 