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

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Abstract

We study the representation theory of a conformal net \({\mathcal{A}}\) on S 1 from a K-theoretical point of view using its universal C*-algebra \({C^*(\mathcal{A})}\). We prove that if \({\mathcal{A}}\) satisfies the split property then, for every representation π of \({\mathcal{A}}\) with finite statistical dimension, \({\pi(C^*(\mathcal{A}))}\) is weakly closed and hence a finite direct sum of type I∞ factors. We define the more manageable locally normal universal C*-algebra \({C_{\rm ln}^*(\mathcal{A})}\) as the quotient of \({C^*(\mathcal{A})}\) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if \({\mathcal{A}}\) is completely rational with n sectors, then \({C_{\rm ln}^*(\mathcal{A})}\) is a direct sum of n type I∞ factors. Its ideal \({\mathfrak{K}_\mathcal{A}}\) of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of \({C^*(\mathcal{A})}\) with finite statistical dimension act on \({\mathfrak{K}_\mathcal{A}}\), giving rise to an action of the fusion semiring of DHR sectors on \({K_0(\mathfrak{K}_\mathcal{A})}\). Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.

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Authors and Affiliations

  1. Dipartimento di Economia, Università di Chieti-Pescara “G. d’Annunzio”, Viale Pindaro, 42, 65127, Pescara, Italy

    Sebastiano Carpi

  2. Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione di Matematica, Sapienza Università di Roma, Via A. Scarpa, 16, 00161, Roma, Italy

    Roberto Conti

  3. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133, Roma, Italy

    Robin Hillier

  4. Mathematical Institute, Department of Analysis, Budapest University of Technology & Economics (BME), Müegyetem rk. 3-9, 1111, Budapest, Hungary

    Mihály Weiner

Authors
  1. Sebastiano Carpi
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  2. Roberto Conti
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  3. Robin Hillier
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  4. Mihály Weiner
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Correspondence to Sebastiano Carpi.

Additional information

Communicated by Y. Kawahigashi

Supported in part by the ERC Advanced Grant 227458 “Operator Algebras and Conformal Field Theory”.

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Carpi, S., Conti, R., Hillier, R. et al. Representations of Conformal Nets, Universal C*-Algebras and K-Theory. Commun. Math. Phys. 320, 275–300 (2013). https://doi.org/10.1007/s00220-012-1561-5

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