Harmonic analysis in (UMD)-spaces: Applications to the theory of bases. (English. Russian original) Zbl 0857.46006

Math. Notes 58, No. 6, 1315-1326 (1995); translation from Mat. Zametki 58, No. 6, 890-905 (1995).
Summary: A general method for the construction of bases and unconditional finite-dimensional basis decompositions for spaces with the property of unconditional martingale differences is proposed. The construction makes use of a certain strongly continuous representation of Cantor’s group in these spaces. The results are applied to vector function spaces and symmetric spaces of measurable operators associated with factors of type II.


46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: DOI


[1] A. V. Bukhvalov, ”Continuity of operators in vector function spaces: application to the theory of bases,”Zap. Nauchn. Sem. LOMI,157, 5–22 (1987). · Zbl 0637.46032
[2] E. Berkson, T. A. Gillespie, and P. S. Muhly, ”Generalized analyticity in UMD spaces,”Arkiv for Math.,27, 1–14 (1989). · Zbl 0705.46013
[3] R. E. Edwards,Fourier Series. A Modern Introduction, Vol. 2, New York (1979).
[4] A. A. Saakyan and B. S. Kashin,Orthogonal Series [in Russian], Nauka, Moscow (1984).
[5] J. Lindenstrauss and L. Tzafriri,Classical Banach spaces, Vol. I, Springer-Verlag, New York (1977). · Zbl 0362.46013
[6] H. W. Ellis, ”On the basis problem for vector-valued functional spaces,”Canad. J. Math.,8, 412–422 (1956). · Zbl 0071.32802
[7] S. G. Krein, Y. I. Petunin, and E. M. Semenov,Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978). · Zbl 0499.46044
[8] J. Lindenstrauss and L. Tzafriri,Classical Banach spaces, Vol. II, Springer-Verlag, New York (1979). · Zbl 0403.46022
[9] G. W. Morgentaller, ”On Walsh-Fourier series,”Trans. Amer. Math. Soc.,84, No. 2, 472–507 (1957). · Zbl 0089.27702
[10] E. Berkson, T. A. Gillespie, and P. S. Muhly, ”Théorie spectrale dans les espaces UMD,”C. R. Acad. Sci. Paris,302, 155–158 (1986). · Zbl 0587.47040
[11] F. A. Sukochev and V. I. Chilin, ”Symmetric spaces on the semifinite von Neumann algebras,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],313, No. 4, 811–816 (1990). · Zbl 0751.46039
[12] S. Sakai,C *-algebras and W*-algebras, Springer-Verlag, New York (1971).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.