Extremal structure in operator spaces
HTML articles powered by AMS MathViewer
- by M. Sharir PDF
- Trans. Amer. Math. Soc. 186 (1973), 91-111 Request permission
Abstract:
(a) A characterization of extreme operators (in the unit ball of operators) between ${L^1}$-spaces is given, together with other related properties, (b) A general theorem of Kreĭn-Milman type for the unit ball of operator spaces is proved, and is applied to operators between ${L^1}$-spaces and to operators into C-spaces.References
- R. M. Blumenthal, Joram Lindenstrauss, and R. R. Phelps, Extreme operators into $C(K)$, Pacific J. Math. 15 (1965), 747–756. MR 209862
- J. Dixmier, Sur certains espaces considérés par M. H. Stone, Summa Brasil. Math. 2 (1951), 151–182 (French). MR 48787
- J. Dixmier, Sur un théorème de Banach, Duke Math. J. 15 (1948), 1057–1071 (French). MR 27440
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
- Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
- A. Grothendieck, Une caractérisation vectorielle-métrique des espaces $L^1$, Canadian J. Math. 7 (1955), 552–561 (French). MR 76301, DOI 10.4153/CJM-1955-060-6
- W. B. Johnson, H. P. Rosenthal, and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. MR 280983, DOI 10.1007/BF02771464
- Shizuo Kakutani, Concrete representation of abstract $(L)$-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523–537. MR 4095, DOI 10.2307/1968915
- Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
- Dorothy Maharam, On homogeneous measure algebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 108–111. MR 6595, DOI 10.1073/pnas.28.3.108
- P. D. Morris and R. R. Phelps, Theorems of Krein-Milman type for certain convex sets of operators, Trans. Amer. Math. Soc. 150 (1970), 183–200. MR 262804, DOI 10.1090/S0002-9947-1970-0262804-3
- Haskell P. Rosenthal, On injective Banach spaces and the spaces $L^{\infty }(\mu )$ for finite measure $\mu$, Acta Math. 124 (1970), 205–248. MR 257721, DOI 10.1007/BF02394572
- M. Sharir, Characterization and properties of extreme operators into $C(Y)$, Israel J. Math. 12 (1972), 174–183. MR 317026, DOI 10.1007/BF02764661
- Kondagunta Sundaresan, Extreme points of the unit cell in Lebesgue-Bochner function spaces, Colloq. Math. 22 (1970), 111–119. MR 276753, DOI 10.4064/cm-22-1-111-119
- Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 186 (1973), 91-111
- MSC: Primary 47D20; Secondary 46B05
- DOI: https://doi.org/10.1090/S0002-9947-1973-0333829-7
- MathSciNet review: 0333829