Abstract
We study problems of interpolation of positive linear operators in couples of ordered Banach spaces. From this viewpoint, we study couples of noncommutative spaces L 1, L ∞ associated with weights and traces on von Neumann algebras.
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References
G. Ya. Lozanovskii, “Remark on a Calderon interpolation theorem,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.] 6(4), 89–90 (1972).
B. Z. Vulikh, Introduction to the Theory of Cones in Normed Spaces (Kalinin State Univ., Kalinin, 1977) [in Russian].
M. A. Krasnoselskii, E. A. Lifshits, and A. V. Sobolev, Positive Linear Systems: Method of Positive Operators (Nauka, Moscow, 1985) [in Russian].
L. Asimow and A. J. Ellis, Convexity Theory and Its Applications in Functional Analysis (Academic Press, London, 1980).
Y. A. Abramovich and C. D. Aliprantis, “Positive operators,” in Handbook of the Geometry of Banach Spaces (Elsevier, Amsterdam, 2001), Vol. 1, pp. 85–122.
B. Z. Vulikh, Special Problems of Geometry of Cones in Normed Spaces (Kalinin State Univ., Kalinin, 1978) [in Russian].
S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators (Nauka, Moscow, 1978) [in Russian].
A. V. Bukhvalov, V. B. Korotkov, A. G. Kusraev, S. S. Kutateladze, and B. M. Makarov, Vector Lattices and Integral Operators (Nauka. Siberian Branch, Novosibirsk, 1992) [in Russian].
N. Aronszajn and E. Gagliardo, “Interpolation spaces and interpolation methods,” Ann. Mat. Pura Appl. (4) 68, 51–117 (1965).
Yu. A. Brudnyi, S. G. Krein, and E. M. Semenov, “Interpolation of linear operators,” in Mathematical Analysis, Itogi Nauki i Tekhniki, Vol. 24 (VINITI, Moscow, 1986), pp. 3–163 [in Russian].
V. I. Ovchinnikov, “The method of orbits in interpolation theory,” Math. Rep.(Harwood Academic Publishers) 1(2), 349–515 (1984).
M. Takesaki, Theory of Operator Algebras. I (Springer-Verlag, New York-Heidelberg, 1979); 2nd edition in Encyclopedia of Math. Sci. (Springer-Verlag, Berlin, 2002), Vol. 124.
M. Takesaki, Theory of Operator Algebras. II, in Encyclopedia of Math. Sci. (Springer-Verlag, Berlin, 2003), Vol. 125.
N. V. Trunov and A. N. Sherstnev, “Introduction to the theory of noncommutative integration,” in Current Problems in Mathematics. Newest Results, Itogi Nauki i Tekhniki, Vol. 27 (VINITI, Moscow, 1985), pp. 167–190 [in Russian].
A. N. Sherstnev, “On a noncommutative analog of the space L 1,” in Mathematical Analysis (Kazan State Univ., Kazan, 1978), pp. 112–123 [in Russian].
P. G. Dodds, T. K. Dodds, and B. de Pagter, “Fully symmetric operator spaces,” Integral Equations Operator Theory 15(6), 942–972 (1992).
P. G. Dodds, T. K. Dodds, and B. de Pagter, “Noncommutative Köthe duality,” Trans. Amer. Math. Soc. 339(2), 717–750 (1993).
V. I. Ovchinnikov, “Symmetric spaces of measurable operators,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.] 191(4), 769–771 (1970).
V. I. Chilin, “Triangle inequality in algebras of locally measurable operators,” in Mathematical Analysis and Algebra, Trudy Tashkent Gos. Univ. (Tashkent, 1986), pp. 77–81 [in Russian].
C. A. Akemann, J. Anderson, and G. K. Pedersen, “Triangle inequalities in operator algebras,” Linear and Multilinear Algebra 11(2), 167–178 (1982).
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Original Russian Text © L. V. Veselova, F. A. Sukochev, O. E. Tikhonov, 2007, published in Matematicheskie Zametki, 2007, Vol. 81, No. 1, pp. 43–58.
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Veselova, L.V., Sukochev, F.A. & Tikhonov, O.E. Interpolation of positive operators. Math Notes 81, 37–50 (2007). https://doi.org/10.1134/S000143460701004X
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DOI: https://doi.org/10.1134/S000143460701004X