nLab
C-star-category

Contents

Idea

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

Definition

Definition

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

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

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

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

Remark

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

Examples

Example

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

Example

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

Representation Theory

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

Theorem

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

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