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**Representations of conformal nets, universal C*-algebras and K-theory.** / Carpi, Sebastiano; Conti, Roberto ; Hillier, Robin et al.

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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

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

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

@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",

}

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 -