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

Carpi, Sebastiano and Conti, Roberto and Hillier, Robin and Weiner, Mihály (2013) Representations of conformal nets, universal C*-algebras and K-theory. Communications in Mathematical Physics, 320 (1). pp. 275-300. ISSN 0010-3616

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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.

Item Type:
Journal Article
Journal or Publication Title:
Communications in Mathematical Physics
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/3100/3109
Subjects:
ID Code:
65218
Deposited By:
Deposited On:
12 Jun 2013 09:07
Refereed?:
Yes
Published?:
Published
Last Modified:
01 Jan 2020 08:32