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

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Representations of conformal nets, universal C*-algebras and K-theory. / Carpi, Sebastiano; Conti, Roberto ; Hillier, Robin et al.
In: Communications in Mathematical Physics, Vol. 320, No. 1, 05.2013, p. 275-300.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Carpi, S, Conti, R, Hillier, R & Weiner, M 2013, 'Representations of conformal nets, universal C*-algebras and K-theory', Communications in Mathematical Physics, vol. 320, no. 1, pp. 275-300. https://doi.org/10.1007/s00220-012-1561-5

APA

Carpi, S., Conti, R., Hillier, R., & Weiner, M. (2013). Representations of conformal nets, universal C*-algebras and K-theory. Communications in Mathematical Physics, 320(1), 275-300. https://doi.org/10.1007/s00220-012-1561-5

Vancouver

Carpi S, Conti R, Hillier R, Weiner M. Representations of conformal nets, universal C*-algebras and K-theory. Communications in Mathematical Physics. 2013 May;320(1):275-300. doi: 10.1007/s00220-012-1561-5

Author

Carpi, Sebastiano ; Conti, Roberto ; Hillier, Robin et al. / Representations of conformal nets, universal C*-algebras and K-theory. In: Communications in Mathematical Physics. 2013 ; Vol. 320, No. 1. pp. 275-300.

Bibtex

@article{8f00fd1bf6f24078ae1510202527e1d2,
title = "Representations of conformal nets, universal C*-algebras and K-theory",
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.",
author = "Sebastiano Carpi and Roberto Conti and Robin Hillier and Mih{\'a}ly Weiner",
year = "2013",
month = may,
doi = "10.1007/s00220-012-1561-5",
language = "English",
volume = "320",
pages = "275--300",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer New York",
number = "1",

}

RIS

TY - JOUR

T1 - Representations of conformal nets, universal C*-algebras and K-theory

AU - Carpi, Sebastiano

AU - Conti, Roberto

AU - Hillier, Robin

AU - Weiner, Mihály

PY - 2013/5

Y1 - 2013/5

N2 - 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.

AB - 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.

U2 - 10.1007/s00220-012-1561-5

DO - 10.1007/s00220-012-1561-5

M3 - Journal article

VL - 320

SP - 275

EP - 300

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -