Minimum index for subfactors and entropy. II
ジャーナル
フリー
1991 年 43 巻 2 号 p. 347-379
詳細
- Published: 1991 Received: 1990/06/18 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
-
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報 訂正内容: Wrong : 1) L. Accardi and C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal., 45 (1982), 245-273.
2) H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule, Pacific J. Math., 50 (1974), 309-354.
3) H. Araki, Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci., Kyoto Univ., 11 (1976), 809-833.
4) H. Araki, Relative entropy for states of von Neumann algebras II, Publ. Res. Inst. Math. Sci., Kyoto Univ., 13 (1977), 173-192.
5) M. Baillet, Y. Denizeau and J. F. Havet, Indice d'une esperance conditionnelle, Compositio Math., 66 (1988), 199-236.
6) F. Combes and C. Delaroche, Groupe modulaire d'une espérance conditionnelle dans une algèbre de von Neumann, Bull. Soc. Math. France, 103 (1975), 385-426.
7) A. Connes, On hyperfinite factors of type III0 and Krieger's factors, J. Funct. Anal., 18 (1975), 318-327.
8) A. Connes, Classification of injective factors, cases II1, II∞, IIIλ, λ≠1, Ann. of Math., 104 (1976), 73-115.
9) A. Connes, On the spatial theory of von Neumann algebras, J. Funct. Anal., 35 (1980), 153-164.
10) A. Connes, Entropie de Kolmogoroff-Sinai et mécanique statistique quantique, C. R. Acad. Sci. Paris Sér. I, 301 (1985), 1-6.
11) A. Connes, H. Narnhofer and W. Thirring, Dynamical entropy of C* algebras and von Neumann algebras, Comm. Math. Phys., 112 (1987), 691-719.
12) A. Connes and E. Størmer, Entropy for automorphisms of II1 von Neumann algebras, Acta Math., 134 (1975), 289-306.
13) A. Connes and E. Størmer, Homogeneity of the state space of factors of type III1, J. Funct. Anal., 28 (1978), 187-196.
14) G. A. Elliott and E. J. Woods, The equivalence of various definitions for a properly infinite von Neumann algebra to be approximately finite dimensional, Proc. Amer. Math. Soc., 60 (1976), 175-178.
15) F. M. Goodman, P. de la Harpe and V. F. R. Jones, Coxeter Graphs and Towers of Algebras, Springer-Verlag, New York, 1989.
16) U. Haagerup, The standard form of von Neumann algebras, Math. Scand., 37 (1975), 271-283.
17) U. Haagerup, Operator valued weights in von Neumann algebras, I, II, J. Funct. Anal., 32 (1979), 175-206; 33 (1979), 339-361.
18) F. Hiai, Minimizing indices of conditional expectations onto a subfactor, Publ. Res. Inst. Math. Sci., Kyoto Univ., 24 (1988), 673-678.
19) F. Hiai, Minimum index for subfactors and entropy, J. Operator Theory, to appear.
20) F. Hiai and Y. Nakamura, Closed convex hulls of unitary orbits in von Neumann algebras, Trans. Amer. Math. Soc., 323 (1991), 1-38.
21) F. Hiai, M. Ohya and M. Tsukada, Sufficiency, KMS condition and relative entropy in von Neumann algebras, Pacific J. Math., 96 (1981), 99-109.
22) V. F. R. Jones, Index for subfactors, Invent. Math., 72 (1983), 1-25.
23) S. Kawakami, Some remarks on index and entropy for von Neumann subalgebras, Proc. Japan Acad. Ser. A Math. Sci., 65 (1989), 323-325.
24) H. Kosaki, Relative entropy of states: a variational expression, J. Operator Theory, 16 (1986), 335-348.
25) H. Kosaki, Extension of Jones' theory on index to arbitrary factors, J. Funct. Anal., 66 (1986), 123-140.
26) H. Kosaki and R. Longo, in preparation.
27) R. Longo, Index of subfactors and statistics of quantum fields. I, Comm. Math. Phys., 126 (1989), 217-247.
28) Y. Nakagami and M. Takesaki, Duality for Crossed Products of von Neumann Algebras, Lecture Notes in Math., No. 731, Springer-Verlag, Berlin, 1979.
29) M. Ohya, Entropy transmission in C*-dynamical systems, J. Math. Anal. Appl., 100 (1984), 222-235.
30) D. Petz, Properties of the relative entropy of states of von Neumann algebras, Acta Math. Hungar., 47 (1986), 65-72.
31) D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm. Math. Phys., 105 (1986), 123-131.
32) D. Petz, On certain properties of the relative entropy of states of operator algebras, Math. Z., to appear.
33) M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4), 19 (1986), 57-106.
34) M. Pimsner and S. Popa, Iterating the basic construction, Trans. Amer. Math. Soc., 310 (1988), 127-133.
35) S. Popa, Relative dimension, towers of projections and commuting squares of subfactors, Pacific J. Math., 137 (1989), 181-207.
36) S. Stratila, Modular Theory in Operator Algebras, Editura Academiei and Abacus Press, Tunbridge Wells, 1981.
37) M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal., 9 (1972), 306-321.
38) M. Takesaki, Duality for crossed products and structure of von Neumann algebras of type III, Acta Math., 131 (1973), 249-310.
39) A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Comm. Math, phys., 54 (1977), 21-32.
40) H. Umegaki, Conditional expectation in an operator algebra, Tohoku Math. J., 6 (1954), 177-181.
41) H. Umegaki, Conditional expectation in an operator algebra, IV (entropy and information), Kodai Math. Sem. Rep., 14 (1962), 59-85.
42) Y. Watatani, Index for C*-subalgebras, Mem. Amer. Math. Soc., No. 424, Amer. Math. Soc., 1990.
43) A. Wehrl, General properties of entropy, Rev. Modern Phys., 50 (1978), 221-260.
Right : [1] L. Accardi and C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal., 45 (1982), 245-273.
[2] H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule, Pacific J. Math., 50 (1974), 309-354.
[3] H. Araki, Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci., Kyoto Univ., 11 (1976), 809-833.
[4] H. Araki, Relative entropy for states of von Neumann algebras II, Publ. Res. Inst. Math. Sci., Kyoto Univ., 13 (1977), 173-192.
[5] M. Baillet, Y. Denizeau and J. F. Havet, Indice d'une esperance conditionnelle, Compositio Math., 66 (1988), 199-236.
[6] F. Combes and C. Delaroche, Groupe modulaire d'une espérance conditionnelle dans une algèbre de von Neumann, Bull. Soc. Math. France, 103 (1975), 385-426.
[7] A. Connes, On hyperfinite factors of type III0 and Krieger's factors, J. Funct. Anal., 18 (1975), 318-327.
[8] A. Connes, Classification of injective factors, cases II1, II∞, IIIλ, λ≠1, Ann. of Math., 104 (1976), 73-115.
[9] A. Connes, On the spatial theory of von Neumann algebras, J. Funct. Anal., 35 (1980), 153-164.
[10] A. Connes, Entropie de Kolmogoroff-Sinai et mécanique statistique quantique, C. R. Acad. Sci. Paris Sér. I, 301 (1985), 1-6.
[11] A. Connes, H. Narnhofer and W. Thirring, Dynamical entropy of C* algebras and von Neumann algebras, Comm. Math. Phys., 112 (1987), 691-719.
[12] A. Connes and E. Størmer, Entropy for automorphisms of II1 von Neumann algebras, Acta Math., 134 (1975), 289-306.
[13] A. Connes and E. Størmer, Homogeneity of the state space of factors of type III1, J. Funct. Anal., 28 (1978), 187-196.
[14] G. A. Elliott and E. J. Woods, The equivalence of various definitions for a properly infinite von Neumann algebra to be approximately finite dimensional, Proc. Amer. Math. Soc., 60 (1976), 175-178.
[15] F. M. Goodman, P. de la Harpe and V. F. R. Jones, Coxeter Graphs and Towers of Algebras, Springer-Verlag, New York, 1989.
[16] U. Haagerup, The standard form of von Neumann algebras, Math. Scand., 37 (1975), 271-283.
[17] U. Haagerup, Operator valued weights in von Neumann algebras, I, II, J. Funct. Anal., 32 (1979), 175-206; 33 (1979), 339-361.
[18] F. Hiai, Minimizing indices of conditional expectations onto a subfactor, Publ. Res. Inst. Math. Sci., Kyoto Univ., 24 (1988), 673-678.
[19] F. Hiai, Minimum index for subfactors and entropy, J. Operator Theory, to appear.
[20] F. Hiai and Y. Nakamura, Closed convex hulls of unitary orbits in von Neumann algebras, Trans. Amer. Math. Soc., 323 (1991), 1-38.
[21] F. Hiai, M. Ohya and M. Tsukada, Sufficiency, KMS condition and relative entropy in von Neumann algebras, Pacific J. Math., 96 (1981), 99-109.
[22] V. F. R. Jones, Index for subfactors, Invent. Math., 72 (1983), 1-25.
[23] S. Kawakami, Some remarks on index and entropy for von Neumann subalgebras, Proc. Japan Acad. Ser. A Math. Sci., 65 (1989), 323-325.
[24] H. Kosaki, Relative entropy of states: a variational expression, J. Operator Theory, 16 (1986), 335-348.
[25] H. Kosaki, Extension of Jones' theory on index to arbitrary factors, J. Funct. Anal., 66 (1986), 123-140.
[26] H. Kosaki and R. Longo, in preparation.
[27] R. Longo, Index of subfactors and statistics of quantum fields. I, Comm. Math. Phys., 126 (1989), 217-247.
[28] Y. Nakagami and M. Takesaki, Duality for Crossed Products of von Neumann Algebras, Lecture Notes in Math., No. 731, Springer-Verlag, Berlin, 1979.
[29] M. Ohya, Entropy transmission in C*-dynamical systems, J. Math. Anal. Appl., 100 (1984), 222-235.
[30] D. Petz, Properties of the relative entropy of states of von Neumann algebras, Acta Math. Hungar., 47 (1986), 65-72.
[31] D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm. Math. Phys., 105 (1986), 123-131.
[32] D. Petz, On certain properties of the relative entropy of states of operator algebras, Math. Z., to appear.
[33] M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4), 19 (1986), 57-106.
[34] M. Pimsner and S. Popa, Iterating the basic construction, Trans. Amer. Math. Soc., 310 (1988), 127-133.
[35] S. Popa, Relative dimension, towers of projections and commuting squares of subfactors, Pacific J. Math., 137 (1989), 181-207.
[36] S. Stratila, Modular Theory in Operator Algebras, Editura Academiei and Abacus Press, Tunbridge Wells, 1981.
[37] M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal., 9 (1972), 306-321.
[38] M. Takesaki, Duality for crossed products and structure of von Neumann algebras of type III, Acta Math., 131 (1973), 249-310.
[39] A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Comm. Math, phys., 54 (1977), 21-32.
[40] H. Umegaki, Conditional expectation in an operator algebra, Tôhoku Math. J., 6 (1954), 177-181.
[41] H. Umegaki, Conditional expectation in an operator algebra, IV (entropy and information), Kodai Math. Sem. Rep., 14 (1962), 59-85.
[42] Y. Watatani, Index for C*-subalgebras, Mem. Amer. Math. Soc., No. 424, Amer. Math. Soc., 1990.
[43] A. Wehrl, General properties of entropy, Rev. Modern Phys., 50 (1978), 221-260.
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