×

Non-commutative Banach function spaces. (English) Zbl 0653.46061

A singular value inequality of Markus for compact operators is extended to arbitrary measurable operators affiliated with a semifinite von Neumann algebra and used as a key tool to give a direct and simple proof of the triangle inequality and norm completeness of a general class of non-commutative function spaces.
Reviewer: P.G.Dodds

MSC:

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [CR] Chong, K.M., Rice, N.M.: Equimeasurable rearrangements of functions. Queen’s Papers in Pure and Applied Mathematics, no. 28, Queen’s University, Kingston (1971)
[2] [D] Dixmier, J.: Von Neumann Algebras. Mathematical Libary, vol. 27. Amsterdam: North Holland 1981 · Zbl 0473.46040
[3] [DP] Dodds, P.G., Pagter, B. de: Non-commutative Banach function spaces and their duals, Semester Bericht Functionanalysis, Tübingen, Wintersemester, 1988
[4] [F] Fack, T.: Sur la notion de valeur caractéristique. J. Oper. Theory7, 307-333 (1982) · Zbl 0493.46052
[5] [FK] Fack, T., Kosaki, H.: Generalizeds-numbers of ?-measurable operators. Pac. J. Math.123, 269-300 (1986) · Zbl 0617.46063
[6] [GK] Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear non-selfadjoint operators, Translations of Mathematical Monographs, vol. 18, AMS (1969) · Zbl 0181.13503
[7] [G] Grothendieck, A.: Réarrangements de fonctions et inégalités de convexité dans les algèbres de von Neumann munies d’une trace, Séminaire Bourbaki (1955), 113-01-113-13
[8] [HN] Hiai, F., Nakamura, Y.: Majorizations for generalizeds-numbers in semifinite von Neumann algebras. Math. Z.195, 17-27 (1987) · Zbl 0598.46039 · doi:10.1007/BF01161595
[9] [K] Kosaki, H.: Non-commutative Lorentz spaces associated with a semi-finite von Neumann algebra and applications. Proc. Jap. Acad., Ser. A57, 303-306 (1981) · Zbl 0491.46052 · doi:10.3792/pjaa.57.303
[10] [KPS] Krein, S.G., Petunin, Ju.I., Semenov, E.M.: Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, AMS (1982)
[11] [KR] Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras, vol. I. New York: Academic Press 1983 · Zbl 0518.46046
[12] [L] Luxemburg, W.A.J.: Rearrangement invariant Banach function spaces, Queen’s Papers in Pure and Applied Mathematics, no. 10, 83-144 (1967)
[13] [LS] Lorentz, G.G., Shimogaki, T.: Interpolation theorems for operators in function spaces. J. Funct. Anal.2, 31-51 (1968) · Zbl 0162.44504 · doi:10.1016/0022-1236(68)90024-4
[14] [M] Markus, A.S.: The eigen- and singular values of the sum and product of linear operators. Russ. Math. Surv.19, 91-120 (1964) · Zbl 0133.07205 · doi:10.1070/RM1964v019n04ABEH001154
[15] [N] Nelson, E.: Notes on non-commutative integration. J. Funct. Anal.15, 103-116 (1974) · Zbl 0292.46030 · doi:10.1016/0022-1236(74)90014-7
[16] [P] Petz, D.: Spectral scale of self-adjoint operators and trace inequalities. J. Math. Anal. Appl.109, 74-82 (1985) · Zbl 0655.47032 · doi:10.1016/0022-247X(85)90176-3
[17] [T] Terp, M.:L p -spaces associated with von Neumann algebras, Notes, Copenhagen Univ. (1981)
[18] [Y1] Yeadon, F.J.: Non-commutativeL p >-spaces. Math. Proc. Camb. Philos. Soc.77, 91-102 (1975) · Zbl 0327.46068 · doi:10.1017/S0305004100049434
[19] [Y2] Yeadon, F.J.: Ergodic theorems for semifinite von Neumann algebras: II, Math. Proc. Camb. Philos. Soc.88, 135-147 (1980) · Zbl 0466.46056 · doi:10.1017/S0305004100057418
[20] [Z1] Zaanen, A.C.: Integration. Amsterdam: North-Holland 1967
[21] [Z2] Zaanen, A.C.: Riesz spaces II. Amsterdam: North-Holland 1983 · Zbl 0519.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.