Commuting squares and a new relative entropy

Yuki SEO

doi:10.2969/jmsj/04920329

Yuki SEO

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ジャーナルフリー

1997 年 49 巻 2 号 p. 329-337

DOI https://doi.org/10.2969/jmsj/04920329

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Published: 1997 Received: 1994/08/25 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) V. P. Belavkin and P. Staszewski, C^*-algebraic generalization of relative entropy and entropy, Ann. Inst. H. Poincaré, 37 (1982), 51-58.
2) P. Bilingsly, Ergodic Theory and Information, Wiley Sons, 1965.
3) A. Connes and E. Størmer, Entropy for automorphisms of II₁ von Neumann algebras,Acta. Math., 134 (1975), 288-306.
4) J. I. Fujii, M. Fujii and Y. Seo, An extension of the Kubo-Ando theory: Solidarities, Math. Japon., 35 (1990), 387-396.
5) J. I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon., 34 (1989), 341-348.
6) F. Goodman, P. de la Harpe and V. F. R. Jones, Coxeter Graphs and Towers of Algebras, Springer-Verlag, 1989.
7) F. Hiai, Minimum index for subfactors and entropy, J. Operator Theory, 24 (1990), 301-336.
8) F. Hiai, Minimum index for subfactors and entropy, II, J. Math Soc. Japan, 43 (1991), 347-379.
9) F. Hiai, Relative Hamiltonian and relative entropy, preliminary version.
10) F. Hiai and D. Petz, The proper formula for relative entropy and asymptotics in quantum probability, Comm. Math. Phys., 143 (1991), 99-114.
11) V. F. R. Jones, Index for subfactors, Invent. Math., 72 (1983), 1-25.
12) M. Nakamura and H. Umegaki, A note on the entropy for operator algebras, Proc. Japan Acad., 37 (1961), 149-154.
13) M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup., 19 (1986), 57-106.
14) T. Sano and Y. Watatani, Angles between two subfactors, J. Operator Theory, 32 (1994), 209-241.
15) Y. Seo, A new relative entropy for von Neumann algebras, Math. Japon., 36 (1991), 51-57.
16) Y. Seo, Relative entropy for finite von Neumann algebras, J. Operator Theory, 29 (1993), 373-387.
17) M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979.
18) H. Umegaki, Conditional expectation in an operator algebra, Kôdai Math. Sem. Rep., 14 (1962), 59-85.
19) H. Umegaki and M. Ohya, Quantum Mechanical Entropies (in Japanese), Kyoritsu Publishing Company, 1984.
20) Y. Watatani and J. Wierzbicki, Commuting square and relative entropy for two subfactors, to appear in J. Funct. Anal.

Right : [1] V. P. Belavkin and P. Staszewski, C^*-algebraic generalization of relative entropy and entropy, Ann. Inst. H. Poincaré, 37 (1982), 51-58.
[2] P. Bilingsly, Ergodic Theory and Information, Wiley Sons, 1965.
[3] A. Connes and E. Størmer, Entropy for automorphisms of II₁ von Neumann algebras, Acta. Math., 134 (1975), 288-306.
[4] J. I. Fujii, M. Fujii and Y. Seo, An extension of the Kubo-Ando theory: Solidarities, Math. Japon., 35 (1990), 387-396.
[5] J. I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon., 34 (1989), 341-348.
[6] F. Goodman, P. de la Harpe and V. F. R. Jones, Coxeter Graphs and Towers of Algebras, Springer-Verlag, 1989.
[7] F. Hiai, Minimum index for subfactors and entropy, J. Operator Theory, 24 (1990), 301-336.
[8] F. Hiai, Minimum index for subfactors and entropy, II, J. Math Soc. Japan, 43 (1991), 347-379.
[9] F. Hiai, Relative Hamiltonian and relative entropy, preliminary version.
[10] F. Hiai and D. Petz, The proper formula for relative entropy and asymptotics in quantum probability, Comm. Math. Phys., 143 (1991), 99-114.
[11] V. F. R. Jones, Index for subfactors, Invent. Math., 72 (1983), 1-25.
[12] M. Nakamura and H. Umegaki, A note on the entropy for operator algebras, Proc. Japan Acad., 37 (1961), 149-154.
[13] M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup., 19 (1986), 57-106.
[14] T. Sano and Y. Watatani, Angles between two subfactors, J. Operator Theory, 32 (1994), 209-241.
[15] Y. Seo, A new relative entropy for von Neumann algebras, Math. Japon., 36 (1991), 51-57.
[16] Y. Seo, Relative entropy for finite von Neumann algebras, J. Operator Theory, 29 (1993), 373-387.
[17] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979.
[18] H. Umegaki, Conditional expectation in an operator algebra, Kôdai Math. Sem. Rep., 14 (1962), 59-85.
[19] H. Umegaki and M. Ohya, Quantum Mechanical Entropies (in Japanese), Kyoritsu Publishing Company, 1984.
[20] Y. Watatani and J. Wierzbicki, Commuting square and relative entropy for two subfactors, to appear in J. Funct. Anal..

訂正日: 2006/10/20 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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