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%%% Primary Title: 2-C*-category
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%%% Owner: bci1
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\begin{document}
\begin{definition}
A \emph{$2-C^*$ -category}, ${\mathcal{C}^*}_2$, is defined as a (small)
2-category for which the following conditions hold:
\begin{enumerate}
\item for each pair of $1$-arrows $(\rho, \sigma)$ the space $Hom(\rho, \sigma)$ is a complex \htmladdnormallink{Banach space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html}.
\item there is an anti-linear involution `$*$' acting on $2$-arrows, that is,
$ * : Hom(\rho, \sigma) \to Hom(\rho, \sigma)$, $ S \mapsto S^*$ , with $\rho$ and $\sigma$ being $2$-arrows;
\item the Banach \htmladdnormallink{norm}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} is sub-multiplicative (that is,
$$\left\|T \circ S\right\| \leq \left\|S\right\|\left\|T\right\|$$, when the \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} is defined,
and satisfies the $C^*$ -condition:
$$\left\|S^* \circ S\right\| = \left\|S^2\right\|; $$
\item for any 2-arrow $S \in Hom(\rho, \sigma)$, $S^* \circ S$ is a positive element in
$Hom(\rho, \rho)$, (denoted also as $End(\rho)$).
\end{enumerate}
\end{definition}
\textbf{Note:}
The set of $2$-arrows $End(\iota A)$ is a commutative \htmladdnormallink{monoid}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, with the \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html} map
$\iota : \mathcal{C}^{2*}_0 \to \mathcal{C}^{2*}_1$ assigning to each \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $A \in \mathcal{C}^{2*}_0$ a $1$-arrow $\iota A$ such that $$s(\iota A) = t(\iota A) = A.$$
\end{document}
