finite-dimensional case, we need to drop the assumption of a unit.

[H\*]{}-algebras {#sec:hstar}
================

We begin our investigation of suitable axioms for a notion of algebra which can characterize orthonormal bases in arbitrary dimension by recalling the axioms for Frobenius structures. $$\begin{aligned}
      ({\ensuremath{\mathrm{id}_{A}}} \otimes \delta) {\circ}\delta 
  & = (\delta \otimes {\ensuremath{\mathrm{id}_{A}}}) {\circ}\delta \tag{A} \\
      ({\ensuremath{\mathrm{id}_{A}}} {\ensuremath{\otimes}}\varepsilon) {\circ}\delta 
  & = {\ensuremath{\mathrm{id}_{A}}} \tag{U} \\
      \sigma {\circ}\delta 
  & = \delta \tag{C} \\
      \delta^\dag {\circ}\delta 
  & = {\ensuremath{\mathrm{id}_{A}}} \tag{M} \\
      \delta{\circ}\delta^\dag 
  & = (\delta^\dag {\ensuremath{\otimes}}{\ensuremath{\mathrm{id}_{A}}}) {\circ}({\ensuremath{\mathrm{id}_{A}}} {\ensuremath{\otimes}}\delta) & \tag{F}\end{aligned}$$ We note in passing that there is some redundancy in the definition of Frobenius structure.

In any dagger monoidal category, (M), (F) and (F’) imply (A).

$$\begin{aligned}
    {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/A-lhs.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
    & \; \stackrel{\text{(M)}}{=} \;
    {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/MFimplyA-1.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
    \; \stackrel{\text{(F)}}{=} \;
    {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/MFimplyA-2.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
    \; \stackrel{\text{(F')}}{=} \;
    {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/MFimplyA-3.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}} \\
    & \; \stackrel{\text{(F)}}{=} \;
    {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/MFimplyA-4.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
    \; \stackrel{\text{(F')}}{=} \;
    {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/MFimplyA-5.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
    \; \stackrel{\text{(M)}}{=} \;
    {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/A-rhs.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}
  {\tag*{$\Box$}}\end{aligned}$$

The axioms (U), (C), and (F) are independent:

-   As we have seen, for an orthonormal basis $\{{{|} n\rangle} \mid n \in
       {\ensuremath{\mathbb{N}}}\}$ of a separable (infinite-dimensional) Hilbert space, the map $\delta({{|} n\rangle}) = {{|} nn\rangle}$ satisfies everything 