Denting and strongly extreme points in the unit ball of spaces of operators



For 1 ≤p ≤ ∞ we show that there are no denting points in the unit ball of ℓ(lp). This extends a result recently proved by Grząślewicz and Scherwentke whenp = 2 [GS1]. We also show that for any Banach spaceX and for any measure space (Ω, A, μ), the unit ball of ℓ(L 1 (μ), X) has denting points iffL 1(μ) is finite dimensional and the unit ball ofX has a denting point. We also exhibit other classes of Banach spacesX andY for which the unit ball of ℓ(X, Y) has no denting points. When X* has the extreme point intersection property, we show that all ‘nice’ operators in the unit ball of ℓ(X, Y) are strongly extreme points.


Denting point strongly extreme point M-ideal 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    Abramovich Y, New classes of spaces on which compact operators satisfy the Daugavet equation,J. Operator Theory 25 (1991) 331–345MATHMathSciNetGoogle Scholar
  2. [Ans]
    Ansari S I, On Banach spacesY for whichB(C(Ω), Y) = K(C(Ω),Y), Pac. J. Math. 169 (1995) 201–218MATHMathSciNetGoogle Scholar
  3. [BE]
    Behrends E,M-Structure and the Banach-Stone Theorem, Springer LNM No.736 (1979)Google Scholar
  4. [DU]
    Diestel J and Uhl J J,Vector measures, Math. Surveys, No. 15 (Providence, Rhode Island) (1977)Google Scholar
  5. [DM]
    Diestel J and Morrison T J, The Radon-Nikodym property for the space of operators,Math. Nach. 92 (1979) 7–12MATHCrossRefMathSciNetGoogle Scholar
  6. [GR]
    Grzaslewicz R, A note on extreme contractions on lp spaces,Port. Math. 40 (1981) 413–419MathSciNetGoogle Scholar
  7. [GS1]
    Grząślewicz R and Scherwentke P, On strongly extreme and denting points in ℓ(H),Math. Jpn. 41 (1995) 283–284Google Scholar
  8. [GS1]
    Grząślewicz R and Scherwentke P, On strongly extreme and denting contractions inℓ(C(X), C(Y)), Bull. Acad. Sinica 25 (1997) 155–160Google Scholar
  9. [HWW]
    Harmand P, Werner D and Werner W,M-ideals in Banach spaces and Banach algebras, (Springer LNM No 1547, Berlin) (1993)MATHGoogle Scholar
  10. [HL]
    Hu Z and Lin B L, RNP and CPCP in Lebesgue-Bochner function spaces,Ill. J. Math. 37 (1993) 329–347MATHMathSciNetGoogle Scholar
  11. [H]
    Hennefeld J, Compact extremal operators,Il. J. Math. 21 (1977) 61–65MATHMathSciNetGoogle Scholar
  12. [KW]
    Kalton N J and Werner D, Property (M), M-ideals, and almost isometric structure of Banach spaces,J. Reine Angew. Math. 461 (1995) 137–178MATHMathSciNetGoogle Scholar
  13. [KR]
    Kunen K and Rosenthal H P, Martingale proofs of some geometrical results in Banach space theory,Pac. J. Math. 100 (1982) 153–175MATHMathSciNetGoogle Scholar
  14. [La]
    Lau A T M and Mah P F, Quasinormal structures for certain spaces of operators on a Hubert space,Pac. J. Math. 121 (1986) 109–118MATHMathSciNetGoogle Scholar
  15. [L1]
    Lima Å, Intersection properties of balls in spaces of compact operators,Ann. Inst. Fourier,28(1978)35–65.MATHMathSciNetGoogle Scholar
  16. [L2]
    Lima A, Oja E, Rao T S S R K and Werner D, Geometry of operator spaces,Mich. Math. J. 41 (1994) 473–490MATHCrossRefMathSciNetGoogle Scholar
  17. [LLT]
    Bor-Luh Lin, Pei-Kee Lin and Troyanski S L, Characterizations of denting points,Proc. Amer. Math. Soc. 102 (1988) 526–528MATHCrossRefMathSciNetGoogle Scholar
  18. [LT]
    Lindenstrauss J and Tzafriri L,Classical Banach spaces I, (Springer, Berlin) (1977)MATHGoogle Scholar
  19. [RS]
    Ruess W M and Stegall C P, Weak*-denting points in duals of operator spaces, in “Banach spaces”, ed.: Kalton N and Saab E,Lecture Notes in Mathematics, No. 1166, Springer (1984) 158–168Google Scholar
  20. [R1]
    Rao T S S R K, On the extreme point intersection property, in “Function Spaces, the second conference”,Lecture Notes in pure and applied mathematics Ed.: K. Jarosz, (No. 172, Marcel Dekker) (1995) 339–346Google Scholar
  21. [R2]
    Rao T S S R K, There are no denting points in the unit ball ofWC(K, X), Proc. Am. Math. Soc. (to appear)Google Scholar
  22. [R3]
    Rao TSSRK, Points of weak*-norm continuity in the unit ball of the spaceWC(K, X)*, Can. Math. Bull. (to appear)Google Scholar
  23. [SM]
    Sharir M, Extremal structures in operator spaces,Trans. Am. Math. Soc. 186 (1973) 91–111CrossRefMathSciNetGoogle Scholar
  24. [SuK]
    Sundaresan K, Extreme points of the unit cell in Lebesgue-Bochner function spaces,Colloq. Math. 22 (1970) 111–119MATHMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 1999

Authors and Affiliations

  1. 1.Indian Statistical InstituteBangaloreIndia

Personalised recommendations