category theory

# Contents

## Idea

A $C^*$-category can be thought of as a horizontal categorification of a $C^*$-algebra. Equivalently, a $C^*$-algebra $A$ is thought of as a pointed one-object $C^*$-category $\mathbf{B}A$ (the delooping of $A$). Accordingly, a more systematic name for $C^*$-categories would be $C^*$-algebroids.

## Definition

###### Definition

A $C^*$-category is a $*$-category enriched in the category Ban of Banach spaces such that:

1. Every arrow $a \in Hom(x,y)$ satisfies the $C^*$-identity ${\|a^* a\|} = {\|a\|}^2$.

2. Composition satisfies ${\|{b a}\|} \leq {\|b\|} {\|a\|}$ for all composable pairs of arrows $a$ and $b$. (That is, we give $Ban$ the projective tensor product.)

3. For every arrow $a \in Hom(x,y)$ there exists an arrow $b \in Hom(x,x)$ such that $a^\ast a = b^ \ast b$.

###### Remark

Condition (3) above is equivalent to requiring that every arrow of the form $x^* x$ is positive in the sense of $C^*$-algebras. Unlike $C^*$-algebras, this does not follow automatically, as can be seen by considering the category with two objects $x,y$ with all morphism sets a copy of $\mathbb{C}$ and with involution defined on $a \in Hom(x,y)$ by $a^* = \overline{a}$ if $x=y$ and $a^* = -\overline{a}$ otherwise.

## Examples

###### Example

The $C^\ast$-representation category of a weak Hopf $C^\ast$-algebra (see there for details) is naturally a rigid monoidal $C^\ast$-category.

###### Example

The category $Hilb$ of Hilbert spaces and bounded linear maps is a $C^*$-category.

## Representation Theory

$C^*$-algebras can be represented as algebras of bounded linear operators on some choice of Hilbert space, using the G.N.S. construction. $C^*$-categories have an analogue of the G.N.S. construction that allows them to represented on the category $Hilb$ of Hilbert spaces and bounded linear maps.

###### Theorem

For any (small) $C^*$-category $\mathcal{C}$ there exists a faithful $*$-functor $\rho \colon \mathcal{C} \to Hilb$.

Revised on February 27, 2014 03:12:26 by davidoslive? (143.167.5.227)