2CCategory

Definition 0.1

A $2-C^*$ -category, ${\mathcal{C}^*}_2$ , is defined as a (small) 2-category for which the following conditions hold:

for each pair of -arrows $(\rho, \sigma)$ the space $Hom(\rho, \sigma)$ is a complex Banach space.
there is an anti-linear involution `' acting on -arrows, that is, $* : Hom(\rho, \sigma) \to Hom(\rho, \sigma)$ , $S \mapsto S^*$ , with $\rho$ and $\sigma$ being -arrows;
the Banach norm is sub-multiplicative (that is,

$\displaystyle \left\Vert T \circ S\right\Vert \leq \left\Vert S\right\Vert\left\Vert T\right\Vert$
, when the composition is defined, and satisfies the -condition:

$\displaystyle \left\Vert S^* \circ S\right\Vert = \left\Vert S^2\right\Vert;$
for any 2-arrow $S \in Hom(\rho, \sigma)$ , $S^* \circ S$ is a positive element in $Hom(\rho, \rho)$ , (denoted also as $End(\rho)$ ).

Note: The set of $2$

-arrows $End(\iota A)$ is a commutative monoid, with the identity map $\iota : \mathcal{C}^{2*}_0 \to \mathcal{C}^{2*}_1$ assigning to each object $A \in \mathcal{C}^{2*}_0$ a $1$

-arrow $\iota A$ such that