Abstract
Using fusion rules of sectors as a working hypothesis, we construct endomorphisms of the Cuntz algebra\(\mathcal{O}_n \) whose images have finite Watatani indices. Quasi-free KMS states on\(\mathcal{O}_n \) appear in a natural way associated with the endomorphisms, and we determine the Murray-von Neumann-Connes types of their GNS representations.
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[BE] Bratteli, O., Evans, D.: Derivation tangential to compact group: The non-abelian case. Proc. Lond. Math. Soc.52, 369–384 (1986)
[BR] Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics II. Berlin, Heidelberg, New York: Springer 1981
[C1] Cuntz, J.: Simple C*-algebras generated by isometries. Commun. Math. Phys.57, 173–185 (1977)
[C2] Cuntz, J.: Regular actions of Hopf algebras on the C*-algebra generated by a Hilbert space. Preprint
[Co] Connes, A.: Periodic automorphisms of the hyperfinite factor of type II1. Acta. Sci. Math.39, 39–66 (1977)
[CDPR] Ceccherini, T., Roberts, J.E., Doplicher, S., Pinzari, C.: in preparation
[CK] Choda, M., Kosaki, H.: Strongly outer actions for an inclusion of factors. Preprint
[DHR] Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics. I. Commun. Math. Phys.35, 199–230 (1971)
[DR] Doplicher, S., Roberts, J.E.: Duals of Lie groups realized in the Cuntz algebras and their actions on C*-algebras. J. Funct. Anal.74, 96–120 (1987)
[FRS] Fredenhagen, K., Rehren, K.H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. I: General theory. Commun. Math. Phys.125, 201–226 (1989)
[GHJ] Goodman, F., de la Harpe, P., Jones, V.: Coxeter graphs and towers of algebras. MSRI Publications14, Berlin, Heidelberg, New York: Springer 1989
[H] Hiai, F.: Minimizing indices of conditional expectations on a subfactor. Publ. RIMS, Kyoto Univ.24, 673–678 (1988)
[I1] Izumi, M.: Application of fusion rules to classification of subfactors. Publ. RIMS, Kyoto Univ.27, 953–994 (1991)
[I2] Izumi, M.: Goldman's type theorem for index 3. To appear in Publ. RIMS, Kyoto Univ.
[I3] Izumi, M.: On flatness of the Coxeter graphE 8. To appear in Pacific J. Math.
[I4] Izumi, M.: On type II and type III principal graphs of subfactors. To appear in Math. Scand.
[I5] Izumi, M.: Subalgebras of infinite C*-algebras with finite Watatani indices. II. Cuntz-Krieger algebras. In preparation
[IK] Izumi, M., Kawahigashi, Y.: Classification of subfactors with the principal graphD 1 n . To appear in J. Funct. Anal.
[J] Jones, V.: Index for subfactors. Invent. Math.72, 1–25 (1983)
[K] Kosaki, H.: Extension of Jones theory on index to arbitrary factors. J. Funct. Anal.66, 123–140 (1986)
[KL] Kosaki, H., Longo, R.: A remark on the minimal index of subfactors. J. Funct. Anal.107, 458–470 (1992)
[Ka] Kawahigashi, Y.: On flatness of Ocneanu's connections on the Dynkin diagrams and classification of subfactors. Preprint
[L1] Longo, R.: Index of subfactors and statistics of quantum fields I. Commun. Math. Phys.126, 217–247 (1989)
[L2] Longo, R.: Index of subfactors and statistics of quantum fields II. Commun. Math. Phys.130, 285–309 (1990)
[L3] Longo, R.: Simple injective subfactors. Adv. Math.63, 152–171 (1987)
[L4] Longo, R.: Minimal index and braided subfactors. J. Funct. Anal.109, 98–112 (1992)
[L5] Longo, R.: A duality for Hopf algebras and for subfactors. Preprint
[O1] Ocneanu, A.: Quantized group string algebra and Galois theory for algebra. In: Operator algebras and applications, Vol. 2 (Warwick, 1987), London Math. Soc. Lect. Note Series Vol.136, Cambridge: Cambridge University Press, 1988, pp. 119–172
[O2] Ocneanu, A.: Graph geometry, quantized group and amenable subfactors. Lake Tahoe Lectures, June–July, 1989
[O3] Ocneanu, A.: Quantum symmetry, differential geometry of finite graphs and classification of subfactors. University of Tokyo Seminary Notes, (Notes recorded by Y. Kawahigashi) 1990
[P] Pinzari, C.: Private communication
[PP] Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Scient. Éc. Norm. Sup.4, 57–106 (1986)
[S] Strătilă, S.: Modular theory in operator algebras. Editra Academiei and Abacus Press 1981
[SV] Sunder, S.V., Vijayarajan, A.K.: On the non-occurrence of the Coxeter graphsE 7,D odd as principal graphs of an inclusion of II1 factors. To appear in Pac. J. Math.
[W] Watatani, Y.: Index for C*-algebras. Mem. Am. Math. Soc.83, No. 424 (1990)
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Communicated by A. Jaffe
Dedicated to Huzihiro Araki
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Izumi, M. Subalgebras of infinite C*-algebras with finite Watatani indices I. Cuntz algebras. Commun.Math. Phys. 155, 157–182 (1993). https://doi.org/10.1007/BF02100056
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DOI: https://doi.org/10.1007/BF02100056