Subalgebras of infinite $C^\ast$-algebras with finite Watatani indices, II: Cuntz-Krieger algebras

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Subalgebras of infinite $C^\ast$-algebras with finite Watatani indices, II: Cuntz-Krieger algebras

Masaki Izumi

Source: Duke Math. J. Volume 91, Number 3 (1998), 409-461.

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See also: Masaki Izumi. Subalgebras of infinite C∗-algebras with finite Watatani indices, I: Cuntz algebras. Comm. Math. Phys. Vol. 155 (1993), pp. 157–182.

Mathematical Reviews (MathSciNet): MR1228532

Primary Subjects: 46L05

Secondary Subjects: 46L37

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References

[BS] S. Baaj and G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $C\sp *$-algèbres, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 425–488.

Mathematical Reviews (MathSciNet): MR94e:46127

Zentralblatt MATH: 0804.46078

[BB] E. Bannai and E. Bannai, Spin models on finite cyclic groups, J. Algebraic Combin. 3 (1994), no. 3, 243–259.

Mathematical Reviews (MathSciNet): MR95j:05173

Zentralblatt MATH: 0803.05058

Digital Object Identifier: doi:10.1023/A:1022407800541

[BR] O. Bratteli and D. W. Robinson, Operator algebras and quantum-statistical mechanics. II, Springer-Verlag, New York, 1981.

Mathematical Reviews (MathSciNet): MR82k:82013

Zentralblatt MATH: 0463.46052

[CDPR] T. Ceccherini, S. Doplicher, C. Pinzari, and J. E. Roberts, A generalization of the Cuntz algebras and model actions, J. Funct. Anal. 125 (1994), no. 2, 416–437.

Mathematical Reviews (MathSciNet): MR96e:46071

Zentralblatt MATH: 0816.46068

Digital Object Identifier: doi:10.1006/jfan.1994.1131

[Ch1] M. Choda, Extension algebras via $\sp *$-endomorphisms, Subfactors (Kyuzeso, 1993), World Sci. Publishing, River Edge, NJ, 1994, pp. 105–128.

Mathematical Reviews (MathSciNet): MR96a:46106

Zentralblatt MATH: 0927.46041

[Ch2] M. Choda, Square roots of the canonical shifts, J. Operator Theory 31 (1994), no. 1, 145–163.

Mathematical Reviews (MathSciNet): MR96b:46086

Zentralblatt MATH: 0832.46057

[Cu1] J. Cuntz, Simple $C\sp*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185.

Mathematical Reviews (MathSciNet): MR57:7189

Zentralblatt MATH: 0399.46045

Digital Object Identifier: doi:10.1007/BF01625776

Project Euclid: euclid.cmp/1103901288

[Cu2] J. Cuntz, $K$-theory for certain $C\sp\ast$-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197.

Mathematical Reviews (MathSciNet): MR84c:46058

Zentralblatt MATH: 0437.46060

Digital Object Identifier: doi:10.2307/1971137

JSTOR: links.jstor.org

[Cu3] J. Cuntz, Regular actions of Hopf algebras on the $C\sp \ast$-algebra generated by a Hilbert space, Operator algebras, mathematical physics, and low-dimensional topology (Istanbul, 1991), Res. Notes Math., vol. 5, A K Peters, Wellesley, MA, 1993, pp. 87–100.

Mathematical Reviews (MathSciNet): MR94m:46104

Zentralblatt MATH: 0803.46062

[CK] J. Cuntz and W. Krieger, A class of $C\sp\ast$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268.

Mathematical Reviews (MathSciNet): MR82f:46073a

Zentralblatt MATH: 0434.46045

Digital Object Identifier: doi:10.1007/BF01390048

[D] M.-C. David, Paragroupe d'Adrian Ocneanu et algèbre de Kac, Pacific J. Math. 172 (1996), no. 2, 331–363.

Mathematical Reviews (MathSciNet): MR98f:46051

Zentralblatt MATH: 0852.46054

Project Euclid: euclid.pjm/1102366014

[DR] S. Doplicher and J. E. Roberts, Duals of compact Lie groups realized in the Cuntz algebras and their actions on $C\sp \ast$-algebras, J. Funct. Anal. 74 (1987), no. 1, 96–120.

Mathematical Reviews (MathSciNet): MR89a:22011

Zentralblatt MATH: 0619.46053

Digital Object Identifier: doi:10.1016/0022-1236(87)90040-1

[EN] M. Enock and R. Nest, Irreducible inclusions of factors, multiplicative unitaries, and Kac algebras, J. Funct. Anal. 137 (1996), no. 2, 466–543.

Mathematical Reviews (MathSciNet): MR97f:46098

Zentralblatt MATH: 0847.22003

Digital Object Identifier: doi:10.1006/jfan.1996.0053

[ES] M. Enock and J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer-Verlag, Berlin, 1992.

Mathematical Reviews (MathSciNet): MR94e:46001

Zentralblatt MATH: 0805.22003

[EK] D. E. Evans and Y. Kawahigashi, Orbifold subfactors from Hecke algebras, Comm. Math. Phys. 165 (1994), no. 3, 445–484.

Mathematical Reviews (MathSciNet): MR96c:46059

Zentralblatt MATH: 0805.46077

Digital Object Identifier: doi:10.1007/BF02099420

Project Euclid: euclid.cmp/1104271410

[FRS] K. Fredenhagen, K.-H. Rehren, and B. Schroer, Superselection sectors with braid group statistics and exchange algebras. I. General theory, Comm. Math. Phys. 125 (1989), no. 2, 201–226.

Mathematical Reviews (MathSciNet): MR91c:81047

Zentralblatt MATH: 0682.46051

Digital Object Identifier: doi:10.1007/BF01217906

Project Euclid: euclid.cmp/1104179464

[GJ] D. M. Goldschmidt and V. F. R. Jones, Metaplectic link invariants, Geom. Dedicata 31 (1989), no. 2, 165–191.

Mathematical Reviews (MathSciNet): MR91e:57014

Zentralblatt MATH: 0678.57007

Digital Object Identifier: doi:10.1007/BF00147477

[GHJ] F. Goodman, P. de la Harpe, and V. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989.

Mathematical Reviews (MathSciNet): MR91c:46082

Zentralblatt MATH: 0698.46050

[HO] R. Herman and A. Ocneanu, Index theory and Galois theory for infinite index inclusions of factors, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 17, 923–927.

Mathematical Reviews (MathSciNet): MR91e:46082

Zentralblatt MATH: 0692.46050

[H] F. Hiai, Minimizing indices of conditional expectations onto a subfactor, Publ. Res. Inst. Math. Sci. 24 (1988), no. 4, 673–678.

Mathematical Reviews (MathSciNet): MR90a:46157

Zentralblatt MATH: 0679.46050

Digital Object Identifier: doi:10.2977/prims/1195174872

[I1] M. Izumi, Application of fusion rules to classification of subfactors, Publ. Res. Inst. Math. Sci. 27 (1991), no. 6, 953–994.

Mathematical Reviews (MathSciNet): MR93b:46121

Zentralblatt MATH: 0765.46048

Digital Object Identifier: doi:10.2977/prims/1195169007

[I2] M. Izumi, Goldman's type theorem for index $3$, Publ. Res. Inst. Math. Sci. 28 (1992), no. 5, 833–843.

Mathematical Reviews (MathSciNet): MR94m:46093

Zentralblatt MATH: 0809.46071

Digital Object Identifier: doi:10.2977/prims/1195167939

[I3] M. Izumi, Subalgebras of infinite $C\sp *$-algebras with finite Watatani indices. I. Cuntz algebras, Comm. Math. Phys. 155 (1993), no. 1, 157–182.

Mathematical Reviews (MathSciNet): MR94e:46104

Zentralblatt MATH: 0803.46066

Digital Object Identifier: doi:10.1007/BF02100056

Project Euclid: euclid.cmp/1104253206

[I4] M. Izumi, Goldman's type theorems in index theory, in Proc. of Operator Algebras and Quantum Field Theory, to appear.

[IK] M. Izumi and Y. Kawahigashi, Classification of subfactors with the principal graph $D\sp (1)\sb n$, J. Funct. Anal. 112 (1993), no. 2, 257–286.

Mathematical Reviews (MathSciNet): MR94k:46127

Zentralblatt MATH: 0791.46039

Digital Object Identifier: doi:10.1006/jfan.1993.1033

[J] V. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25.

Mathematical Reviews (MathSciNet): MR84d:46097

Zentralblatt MATH: 0508.46040

Digital Object Identifier: doi:10.1007/BF01389127

[KP] G. I. Kac and V. G. Paljutkin, Finite ring groups, Trudy Moskov. Mat. Obšč. 15 (1966), 224–261, trans. in Trans. Moscow Math. Soc. (1966), 251–294.

Mathematical Reviews (MathSciNet): MR34:8211

Zentralblatt MATH: 0218.43005

[KW] T. Kajiwara and Y. Watatani, Jones index theory by Hilbert modules and $K$-theory, preprint, 1995.

[Ka1] Y. Kawahigashi, On flatness of Ocneanu's connections on the Dynkin diagrams and classification of subfactors, J. Funct. Anal. 127 (1995), no. 1, 63–107.

Mathematical Reviews (MathSciNet): MR95j:46075

Zentralblatt MATH: 0829.46048

Digital Object Identifier: doi:10.1006/jfan.1995.1003

[Ka2] Y. Kawahigashi, Classification of paragroup actions in subfactors, Publ. Res. Inst. Math. Sci. 31 (1995), no. 3, 481–517.

Mathematical Reviews (MathSciNet): MR96i:46077

Zentralblatt MATH: 0837.46046

Digital Object Identifier: doi:10.2977/prims/1195164051

[Ki] E. Kirchberg, The classification of purely infinite $C^\ast$-algebras using Kaspalov's theory, preprint.

[Ko] H. Kosaki, Extension of Jones' theory on index to arbitrary factors, J. Funct. Anal. 66 (1986), no. 1, 123–140.

Mathematical Reviews (MathSciNet): MR87g:46093

Zentralblatt MATH: 0607.46034

Digital Object Identifier: doi:10.1016/0022-1236(86)90085-6

[KL] H. Kosaki and R. Longo, A remark on the minimal index of subfactors, J. Funct. Anal. 107 (1992), no. 2, 458–470.

Mathematical Reviews (MathSciNet): MR93i:46108

Zentralblatt MATH: 0791.46040

Digital Object Identifier: doi:10.1016/0022-1236(92)90118-3

[L1] R. Longo, Index of subfactors and statistics of quantum fields. I, Comm. Math. Phys. 126 (1989), no. 2, 217–247.

Mathematical Reviews (MathSciNet): MR91c:46097

Zentralblatt MATH: 0682.46045

Digital Object Identifier: doi:10.1007/BF02125124

Project Euclid: euclid.cmp/1104179850

[L2] R. Longo, Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial, Comm. Math. Phys. 130 (1990), no. 2, 285–309.

Mathematical Reviews (MathSciNet): MR91j:46083

Zentralblatt MATH: 0705.46038

Digital Object Identifier: doi:10.1007/BF02473354

Project Euclid: euclid.cmp/1104200513

[L3] R. Longo, Simple injective subfactors, Adv. in Math. 63 (1987), no. 2, 152–171.

Mathematical Reviews (MathSciNet): MR88i:46082

Zentralblatt MATH: 0616.46054

Digital Object Identifier: doi:10.1016/0001-8708(87)90051-X

[L4] R. Longo, Minimal index and braided subfactors, J. Funct. Anal. 109 (1992), no. 1, 98–112.

Mathematical Reviews (MathSciNet): MR93i:46109

Zentralblatt MATH: 0798.46047

Digital Object Identifier: doi:10.1016/0022-1236(92)90013-9

[L5] R. Longo, A duality for Hopf algebras and for subfactors. I, Comm. Math. Phys. 159 (1994), no. 1, 133–150.

Mathematical Reviews (MathSciNet): MR95h:46097

Zentralblatt MATH: 0802.46075

Digital Object Identifier: doi:10.1007/BF02100488

Project Euclid: euclid.cmp/1104254494

[LR] R. Longo and J. E. Roberts, A theory of dimension, $K$-Theory 11 (1997), no. 2, 103–159.

Mathematical Reviews (MathSciNet): MR98i:46065

Zentralblatt MATH: 0874.18005

Digital Object Identifier: doi:10.1023/A:1007714415067

[MRS] M. H. Mann, I. Raeburn, and C. E. Sutherland, Representations of finite groups and Cuntz-Krieger algebras, Bull. Austral. Math. Soc. 46 (1992), no. 2, 225–243.

Mathematical Reviews (MathSciNet): MR93k:46046

Zentralblatt MATH: 0763.22002

Digital Object Identifier: doi:10.1017/S0004972700011862

[O1] A. Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172.

Mathematical Reviews (MathSciNet): MR91k:46068

Zentralblatt MATH: 0696.46048

[O2] A. Ocneanu, Graph geometry, quantized group, and amenable subfactors, June-July 1989, Lake Tahoe Lectures.

[O3] A. Ocneanu, Quantum symmetry, differential geometry of finite graphs, and classification of subfactors, University of Tokyo Seminar Notes, 1990, (notes recorded by Y. Kawahigashi).

[O4] A. Ocneanu, An invariant coupling between $3$-manifold and subfactors, with connections to topological and conformal quantum field theory, unpublished announcement, 1991.

[Pa] W. Paschke, The crossed product of a $C\sp\ast$-algebra by an endomorphism, Proc. Amer. Math. Soc. 80 (1980), no. 1, 113–118.

Mathematical Reviews (MathSciNet): MR81m:46081

Zentralblatt MATH: 0435.46045

Digital Object Identifier: doi:10.2307/2042155

[PS] D. A. Pask and C. E. Sutherland, Filtered inclusions of path algebras; a combinatorial approach to Doplicher-Roberts duality, J. Operator Theory 31 (1994), no. 1, 99–121.

Mathematical Reviews (MathSciNet): MR96i:46073

Zentralblatt MATH: 0845.46030

[PP] M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106.

Mathematical Reviews (MathSciNet): MR87m:46120

Zentralblatt MATH: 0646.46057

[Pi] C. Pinzari, private communication.

[Po] S. Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163–255.

Mathematical Reviews (MathSciNet): MR95f:46105

Zentralblatt MATH: 0853.46059

Digital Object Identifier: doi:10.1007/BF02392646

[R1] M. Rørdam, Classification of Cuntz-Krieger algebras, $K$-Theory 9 (1995), no. 1, 31–58.

Mathematical Reviews (MathSciNet): MR96k:46103

Zentralblatt MATH: 0826.46064

Digital Object Identifier: doi:10.1007/BF00965458

[R2] M. Rørdam, Classification of certain infinite simple $C\sp *$-algebras, J. Funct. Anal. 131 (1995), no. 2, 415–458.

Mathematical Reviews (MathSciNet): MR96e:46080a

Zentralblatt MATH: 0831.46063

Digital Object Identifier: doi:10.1006/jfan.1995.1095

[S] N. Sato, Fourier transform for paragroups and its application to the depth two case, Publ. Res. Inst. Math. Sci. 33 (1997), no. 2, 189–222.

Mathematical Reviews (MathSciNet): MR99a:46113

Zentralblatt MATH: 0886.46059

Digital Object Identifier: doi:10.2977/prims/1195145447

[Sy] W. Szymański, Finite index subfactors and Hopf algebra crossed products, Proc. Amer. Math. Soc. 120 (1994), no. 2, 519–528.

Mathematical Reviews (MathSciNet): MR94d:46061

Zentralblatt MATH: 0802.46076

Digital Object Identifier: doi:10.2307/2159890

[W] Y. Watatani, Index for $C\sp *$-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no. 424, vi+117.

Mathematical Reviews (MathSciNet): MR90i:46104

Zentralblatt MATH: 0697.46024

[We] H. Wenzl, Hecke algebras of type $A\sb n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349–383.

Mathematical Reviews (MathSciNet): MR90b:46118

Zentralblatt MATH: 0663.46055

Digital Object Identifier: doi:10.1007/BF01404457

[Wo] S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665.

Mathematical Reviews (MathSciNet): MR88m:46079

Zentralblatt MATH: 0627.58034

Digital Object Identifier: doi:10.1007/BF01219077

Project Euclid: euclid.cmp/1104159726

[Y1] S. Yamagami, A note on Ocneanu's approach to Jones' index theory, Internat. J. Math. 4 (1993), no. 5, 859–871.

Mathematical Reviews (MathSciNet): MR95f:46114

Zentralblatt MATH: 0793.46040

Digital Object Identifier: doi:10.1142/S0129167X9300039X

[Y2] S. Yamagami, On Ocneanu's characterization of crossed products, preprint.

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