nLab C-star-category

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 (unital) 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.

Remark

A C*-category can be defined analogously to unital C*-categories, using enriched nonunital categories instead of (unital) enriched categories.

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 $\rho \colon \mathcal{C} \to Hilb$.

Related concepts

• GNS construction

• star-category, dagger-category

• W-star category

• unitary fusion category

• spaceoid

• KK-theory

References

With emphasis on the special case of $W^\ast$-categories:

• P. Ghez, R. Lima, John E. Roberts, Prop. 1.9 in: $W^\ast$-categories, Pacific J. Math. 120 1 (1985) 79-109 [euclid:pjm/1102703884]