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