2-C*-category

Definition 0.1  

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;
  3. 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 $ C^*$ -condition:

    $\displaystyle \left\Vert S^* \circ S\right\Vert = \left\Vert S^2\right\Vert; $

  4. 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

$\displaystyle s(\iota A) = t(\iota A) = A.$



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