Pacific Journal of Mathematics

Distance between unitary orbits in von Neumann algebras.

Fumio Hiai and Yoshihiro Nakamura

Article information

Source
Pacific J. Math., Volume 138, Number 2 (1989), 259-294.

Dates
First available in Project Euclid: 8 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.pjm/1102650151

Mathematical Reviews number (MathSciNet)
MR996202

Zentralblatt MATH identifier
0667.46044

Subjects
Primary: 46L10: General theory of von Neumann algebras
Secondary: 46L50

Citation

Hiai, Fumio; Nakamura, Yoshihiro. Distance between unitary orbits in von Neumann algebras. Pacific J. Math. 138 (1989), no. 2, 259--294. https://projecteuclid.org/euclid.pjm/1102650151


Export citation

References

  • [I] T.Ando, Majorization, doubly stochastic matrices andcomparison ofeigenval- ues,Lecture Notes, Hokkaido Univ., Sapporo, 1982; toappear inLinear Algebra AppL, (1989).
  • [2] T.Ando, Comparisonof norms \\\f(A) - f(B)\\\and \\\f(\A - B\)\\\, Math.Z., 197(1988), 403-409.
  • [3] T. Ando andR. Bhatia, Eigenvalue inequalities associated with the Cartesian decomposition, Linear and Multilinear Algebra, 22 (1987), 133-147.
  • [4] T. Ando andY. Nakamura, Anti-distance between unitary orbits of operators, unpublished notes,1986.
  • [5] 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.
  • [6] E. A. Azoff andC. Davis, On distances between unitary orbits of selfadjoint operators, Acta Sci. Math., 47 (1984),419-439.
  • [7] R.Bhatia and C.Davis,A boundfor thespectral variation of a unitary operator, Linear and Multilinear Algebra, 15 (1984), 71-76.
  • [8] R. Bhatia, C. Davis andP. Koosis, An extremal problem in Fourier analysis with applications to operator theory, J. Funct. Anal., 82 (1989), 138-150.
  • [9] R. Bhatia, C.Davis andA. Mclntosh, Perturbations of spectral subspaces and solution of linear operator equations,Linear Algebra AppL,52/53 (1983), 45-67.
  • [10] R. Bhatia andJ. A. R. Holbrook, Short normal paths andspectral variation, Proc. Amer. Math. So,94 (1985),377-382.
  • [II] K. M.Chong, Some extensions of a theorem of Hardy, Littlewood and Plya and their applications, Canad. J. Math., 26 (1974), 1321-1340.
  • [12] A. Connes, U.Haagerup andE.Stormer, Diameters of state spaces of type III factors, Operator Algebras andtheir Connections with Topology and Ergodic Theory (H. Araki etal. eds.), Lecture Notes inMath., No. 1132,Springer-Verlag, Berlin, 1985, pp. 91-116.
  • [13] A.Connes and E. Stormer, Homogeneity of the state space offactors of type IIIi, J. Funct. Anal., 28 (1978), 187-196.
  • [14] K. R. Davidson, Thedistance between unitary orbits of normal operators,Acta Sci. Math., 50 (1986),213-223.
  • [15] J.Dixmier, Formes lineairessurunanneaud'operateurs, Bull.Soc.Math. France, 81 (1953), 9-39.
  • [16] W. Donoghue, Monotone Matrix Functions andAnalytic Continuation, Springer- Verlag, Berlin-Heidelberg-New York, 1974.
  • [17] T. Fack andH.Kosaki,Generalizeds-numbers of-measurable operators, Pacific J. Math., 123 (1986), 269-300.
  • [18] U. Haagerup, The standard form of von Neumann algebras, Math. Scand., 37 (1975), 271-283.
  • [19] U. Haagerup, LP-spaces associated with an arbitrary von Neumann algebra, Colloq. Internat. CNRS, No.274, 1979, pp. 175-184.
  • [20] P. Hall, On representatives of subsets,J. London Math. Soc, 10 (1935), 26-30.
  • [21] R. H. Herman and A. Ocneanu, Spectral analysis for automorphisms oflJHF C*-algebras, J. Funct. Anal., 66 (1986), 1-10.
  • [22] F. Hiai, Majorization and stochastic maps in von Neumann algebras, J. Math. Anal. Appl., 127 (1987), 18-48.
  • [23] F. Hiai, Spectral relations and unitary mixing in semifinite von Neumannalge- bras, HokkaidoMath. J., 17 (1988), 117-137.
  • [24] F.Hiai and Y. Nakamura, Majorizations for generalized s-numbers in semifinite von Neumann algebras, Math. Z., 195 (1987), 17-27.
  • [25] V. Kaftal and R. Mercer, Spectral projections of L operators in type IIIA von Neumann algebras, Integral Equations Operator Theory,9 (1986),679-693.
  • [26] E. Kamei, Double stochasticity in finite factors, Math. Japon., 29 (1984), 903- 907.
  • [27] E. Kamei,An order on statistical operators implicitly introduced by von Neumann, Math. Japon., 30 (1985), 891-895.
  • [28] F.Kittaneh and H. Kosaki,Inequalitiesfor the Schatten p-norm V, Publ. RIMS, Kyoto Univ., 23 (1987),433-443.
  • [29] H. Kosaki, Applications of uniform convexity of noncommutative LP-spaces, Trans. Amer. Math. Soc, 283 (1984), 265-282.
  • [30] H. Kosaki, On the continuity of the map --> \\from the predual of aW*-algebra, J. Funct. Anal., 59 (1984), 123-131.
  • [31] A. S. Markus, The eigen- and singular values of the sum and product of linear operators, Russian Math. Surveys, 19 (1964), 91-120.
  • [32] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Ap- plications, Academic Press, New York, 1979.
  • [33] Y. Nakamura, An inequality for generalized s-numbers, Integral Equations Op- erator Theory, 10 (1987), 140-145.
  • [34] E. Nelson, Notes on non-commutative integration, J. Funct. Anal., 15 (1974), 103-116.
  • [35] V. I. Ovchinnikov,s-numbers of measurable operators, Functional Anal. Appl., 4(1970), 236-242.
  • [36] D. Petz, Spectral scale of self-adjointoperators and trace inequalities, J. Math. Anal. Appl., 109 (1985), 74-82.
  • [37] R. T. Powers and E. Stormer, Free states of the canonical anticommutation relations, Comm. Math. Phys., 16 (1970), 1-33.
  • [38] Y. Sakai, Weak spectral order of Hardy, Littlewood and Plya, J. Math. Anal. Appl., 108(1985),31-46.
  • [39] I. Segal, A non-commutative extension of abstract integration, Ann. of Math., 57(1953), 401-457.
  • [40] E. M. Stein and G.Weiss, Introduction of Fourier Analysis on EuclideanSpaces, Princeton Univ. Press, Princeton, 1971.
  • [41] V. S. Sunder, Distance between normal operators, Proc. Amer. Math. Soc, 84 (1982), 483-484.
  • [42] M. Terp, LPspaces associated with von Neumann algebras,Notes,Copenhagen Univ., 1981.
  • [43] O. E. Tikhonov, Continuity of operatorfunctions in topologiesconnected with a trace on a von Neumann algebra (Russian), Izv. Vyssh. Uchebn. Zaved. Mat., 1987, no. 1, 77-79; translated in Soviet Math. (Iz. VUZ), 31 (1987), 110-114.
  • [44] H. Umegaki, Conditional expectation in an operator algebra, I, II, III, IV, Thoku Math. J., 6 (1954), 177-181; ibid., 8 (1956), 86-100; Kodai Math. Sem. Rep., 11 (1959), 51-64; ibid., 14 (1962), 59-85.
  • [45] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Dijferentialgleichungen,Math. Ann., 71 (1912), 441-479.
  • [46] F. J. Yeadon, Non-commutative LP-spaces, Math. Proc. Cambridge Philos. Soc, 77 (1975), 91-102.