Banach algebra [@ingelstam:realhstar Corollary 2.2].[^7]

The following lemma establishes *properness*, which corresponds to $x^*$ being the unique vector with the property defining H\*-algebras. It follows that $R(x^*)$ is the adjoint of $R(x)$.

Suppose $\delta \colon A \to A {\ensuremath{\otimes}}A$ in ${{\ensuremath{\mathbf{Hilb}}}}$ satisfies (A) and (H). Then (M) implies properness, [*i.e.* ]{}$aA=0
  \Rightarrow a=0$. Hence (M) holds if and only if the regular representation is monic.

By [@ambrose:hstar Theorem 2.2], $A$ is the direct sum of its trivial ideal $A'$ and a proper H\*-algebra $A''$. Here, the trivial ideal is $A'=\{a \in A \mid aA=0\}$. Since the direct sum of Hilbert spaces is a dagger biproduct, we can write $\delta$ as $\delta' \oplus \delta''
  \colon A \to A {\ensuremath{\otimes}}A$, where $\delta' \colon A' \to A' {\ensuremath{\otimes}}A'$ and $\delta'' \colon A'' \to A'' {\ensuremath{\otimes}}A''$. The latter two morphisms are again dagger monic as a consequence of (M). So the multiplication $\delta'^\dag$ of $A'$ is epic, which forces $A'=0$.

The following proposition summarizes the preceding discussion.

\[prop:hstaralgebra\] Any structure $(A, \mu)$ in ${\ensuremath{\mathbf{Hilb}}}$ satisfying (A), (H) and (M) is an [H\*]{}-algebra (and also satisfies (F)); and conversely, an [H\*]{}-algebra satisfies (A), (H) and (M), and hence also (F).

Ambrose proved a complete structure 