PlanetPhysics/2C Category

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

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

1. for each pair of $1$-arrows $(\rho ,\sigma )$ the space $Hom(\rho ,\sigma )$ is a complex Banach space.
2. 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;

1. the Banach norm is sub-multiplicative (that is,

$\left\|T\circ S\right\|\leq \left\|S\right\|\left\|T\right\|$, when the composition is defined, and satisfies the $C^{*}$ -condition: $\left\|S^{*}\circ S\right\|=\left\|S^{2}\right\|;$

1. 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 $\iota :{\mathcal {C}}_{0}^{2*}\to {\mathcal {C}}_{1}^{2*}$ assigning to each object $A\in {\mathcal {C}}_{0}^{2*}$ a $1$-arrow $\iota A$ such that $s(\iota A)=t(\iota A)=A.$