# Representations of Conformal Nets, Universal C*-Algebras and K-Theory

February 12, 2012

We study the representation theory of a conformal net $A$ on the circle from a
K-theoretical point of view using its universal C*-algebra $C^*(A)$. We prove that
if $A$ satisfies the split property then, for every representation $\pi$ of $A$ with
finite statistical dimension, $\pi(C^*(A))$ is weakly closed and hence a finite
direct sum of type $I_\infty$ factors. We define the more manageable locally
normal universal C*-algebra $C^*_{ln}(A)$ as the quotient of $C^*(A)$ by its largest
ideal vanishing in all locally normal representations and we investigate its
structure. In particular, if $A$ is completely rational with n sectors, then
$C^*_{ln}(A)$ is a direct sum of n type $I_\infty$ factors. Its ideal $K_A$ of compact
operators has nontrivial K-theory, and we prove that the DHR endomorphisms of
$C^*(A)$ with finite statistical dimension act on $K_A$, giving rise to an action of
the fusion semiring of DHR sectors on $K_0(K_A)$. Moreover, we show that this
action corresponds to the regular representation of the associated fusion
algebra.

open access link

to appear in Commun. Math. Phys.

Keywords:

chiral conformal field theory, representations, completely rational nets, universal C*-algebra, K-theory