Pacific Journal of Mathematics

Sufficiency and relative entropy in $\ast $-algebras with applications in quantum systems.

Fumio Hiai, Masanori Ohya, and Makoto Tsukada

Article information

Source
Pacific J. Math., Volume 107, Number 1 (1983), 117-140.

Dates
First available in Project Euclid: 8 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.pjm/1102720743

Mathematical Reviews number (MathSciNet)
MR701812

Zentralblatt MATH identifier
0524.46044

Subjects
Primary: 46L05: General theory of $C^*$-algebras
Secondary: 82A15

Citation

Hiai, Fumio; Ohya, Masanori; Tsukada, Makoto. Sufficiency and relative entropy in $\ast $-algebras with applications in quantum systems. Pacific J. Math. 107 (1983), no. 1, 117--140. https://projecteuclid.org/euclid.pjm/1102720743


Export citation

References

  • [1] H. Araki, Multipletime analyticityof a quantumstatisticalstate satisfying theKMS boundary condition, Publ. RIMS,Kyoto Univ. Ser.A, 4 (1968), 361-371.
  • [2] H. Araki,Someproperties of modular conjugation operator of von Neumannalgebras and a non-commutative Radon-Nikodymtheoremwith a chainrule, Pacific J. Math., 50 (1974), 309-354.
  • [3] H. Araki, On the equivalence of the KMS condition and the variational principlefor quantumlattice systems,Comm. Math. Phys., 38 (1974), 1-10.
  • [4] H. Araki, Relativeentropy of statesof von Neumannalgebras, Publ. RIMS,Kyoto Univ., 11 (1976), 809-833.
  • [5] H. Araki, Relativeentropy for states of von Neumann algebrasII, Publ. RIMS, Kyoto Univ., 13(1977), 173-192.
  • [6] H. Araki and A. Kishimoto, On clustering property, Rep. Math. Phys., 10 (1976), 275-281.
  • [7] R. R. Bahadur, Sufficiencyand statisticaldecision functions, Ann. Math. Statist., 25 (1954), 423-462.
  • [8] O. Bratteli andD.W. Robinson, Operator Algebras andQuantum Statistical Mechanics II, Springer, NewYork,1981.
  • [9] A. Connes, Uneclassification desfacteursde typell, Ann. Sci. Ecole Norm. Sup. Ser. 4,6(1973), 133-252.
  • [10] S. Doplicher, D. Kastler and E. Strmer, Invariant statesandasymptotic abelianness, J. Funct. Anal., 3 (1969), 419-434.
  • [11] S. P. Gudder and R. L. Hudson, A noncommutative probability theory, Trans. Amer. Math. Soc,245 (1978), 1-41.
  • [12] U. Haagerup, Thestandard form of von Neumannalgebras, Math. Scand., 37 (1975), 271-283.
  • [13] P. R. Halmos and L. J. Savage, Application of the Radon-Nikodym theoremto the theoryof sufficient statistics, Ann. Math. Statist., 20 (1949), 225-241.
  • [14] F.Hiai, M.Ohya andM.Tsukada, Sufficiency, KMS condition andrelative entropy in von Neumannalgebras, Pacific J. Math., 96 (1981), 99-109.
  • [15] A. S. Holevo, Problems in the mathematical theoryof quantumcommunication chan- nels,Rep.Math. Phys., 12 (1977), 273-278.
  • [16] A. S. Holevo,Investigations in theGeneral Theory of Statistical Decisions, Amer. Math. Soc, Proc. Steklov Institute of Math., no. 124,1978.
  • [17] R. B. Israel, Convexity in the Theory of Lattice Gases, Princeton Univ. Press, Princeton, 1979.
  • [18] I. Kovacs and J. Szcs, Ergodictype theorems in von Neumann algebras, Acta Sci. Math, 27(1966), 233-246.
  • [19] S. Kullback andR.A. Leibler, On information andsufficiency, Ann.Math. Statist., 22 (1951), 79-86.
  • [20] O. E. Lanford IIIandD.W. Robinson, Statistical mechanics of quantum spin systems. Ill, Comm. Math. Phys, 9 (1968), 327-338.
  • [21] M. Ohya, Quantum ergodic channels in operator algebras, J. Math. Anal. Appl., 84 (1981), 318-327.
  • [22] G. K. Pedersen and M. Takesaki, The Radon-Nikodym theorems for von Neumann algebras, Acta Math., 130 (1973), 53-87.
  • [23] R. T. Powers, Self-adjoint algebras of unbounded operators, Comm. Math. Phys., 21 (1971), 85-124.
  • [24] W. Pusz and S. L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys., 8 (1975), 159-170.
  • [25] D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin, New York-Amsterdam, 1969.
  • [26] D. Ruelle, Integral representation of states on a C*-algebra,J. Functional Anal, 6 (1970), 116-151.
  • [27] E. St0rmer, Large groups of automorphisms of C*'-algebras, Comm. Math. Phys., 5 (1967), 1-22.
  • [28] M. Takesaki, Tomita^s Theory of Modular Hlbert Algebras and its Applications, Springer, Lecture notes in math., Vol. 128, 1970.
  • [29] M. Takesaki, Conditional expectations in von Neumann algebras, J. Functional Anal., 9 (1972), 306-321.
  • [30] M. Takesaki, Theory of Operator Algebras I, Springer, New York, 1979.
  • [31] J. Tomiyama, On the projection of norm one in W*-algebras,Proc. Japan Acad., 33 (1957), 608-612.
  • [32] A. Uhlmann, Relative entropy and the Wigner-anase-Dyson-Lieb concavity in an interpolation theory, Comm. Math. Phys., 54 (1977), 21-32.
  • [33] H. Umegaki, Conditional expectation in an operator algebra, III, Kdai Math. Sem. Rep., 11 (1959), 51-64.
  • [34] H. Umegaki, Conditional expectation in an operator algebra, IV (entropy and information), Kdai Math. Sem. Rep., 14 (1962), 59-85.