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Representations of conformal nets, universal C*-algebras and K-theory

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<mark>Journal publication date</mark>05/2013
<mark>Journal</mark>Communications in Mathematical Physics
Issue number1
Volume320
Number of pages26
Pages (from-to)275-300
Publication StatusPublished
<mark>Original language</mark>English

Abstract

We study the representation theory of a conformal net A on S 1 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 π of A with finite statistical dimension, π(C∗(A)) is weakly closed and hence a finite direct sum of type I∞ 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∞ factors. Its ideal KA of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C∗(A) with finite statistical dimension act on KA , giving rise to an action of the fusion semiring of DHR sectors on K0(KA) . Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.