Uniqueness of norm-preserving extensions of functionals on the space of compact operators

  • Julia Martsinkevitš
  • Märt Põldvere


Godefroy, Kalton, and Saphar called a closed subspace $Y$ of a Banach space $Z$ an ideal if its annihilator $Y^\bot $ is the kernel of a norm-one projection $P$ on the dual space $Z^\ast $. If $Y$ is an ideal in $Z$ with respect to a projection on $Z^\ast $ whose range is norming for $Z$, then $Y$ is said to be a strict ideal. We study uniqueness of norm-preserving extensions of functionals on the space $\mathcal{K}(X,Y) $ of compact operators between Banach spaces $X$ and $Y$ to the larger space $\mathcal{K}(X,Z) $ under the assumption that $Y$ is a strict ideal in $Z$. Our main results are: (1) if $y^\ast $ is an extreme point of $B_{Y^{\ast} }$ having a unique norm-preserving extension to $Z$, and $x^{\ast\ast} \in B_{X^{\ast\ast} }$, then the only norm-preserving extension of the functional $x^{\ast\ast} \otimes y^\ast \in \mathcal {K}(X,Y)^\ast $ to $\mathcal {K}(X,Z)$ is $x^{\ast\ast} \otimes z^\ast $ where $z^\ast \in Z^\ast $ is the only norm-preserving extension of $y^\ast $ to $Z$; (2) if $\mathcal{K}(X,Y) $ is an ideal in $\mathcal{K}(X,Z) $ and $Y$ has Phelps' property $U$ in its bidual $Y^{\ast\ast} $ (i.e., every bounded linear functional on $Y$ admits a unique norm-preserving extension to $Y^{\ast\ast} $), then $\mathcal{K}(X,Y) $ has property $U$ in $\mathcal{K}(X,Z) $ whenever $X^{\ast\ast} $ has the Radon-Nikodým property.


Bourgin, R. D., Geometric aspects of convex sets with the Radon-Nikodým property, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/BFb0069321

Choquet, G., Lectures on analysis. Vol. II: Representation theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

Feder, M. and Saphar, P., Spaces of compact operators and their dual spaces, Israel J. Math. 21 (1975), no. 1, 38–49. https://doi.org/10.1007/BF02757132

Godefroy, G., Points de Namioka. Espaces normants. Applications à la théorie isométrique de la dualité, Israel J. Math. 38 (1981), no. 3, 209–220. https://doi.org/10.1007/BF02760806

Godefroy, G., Kalton, N. J., and Saphar, P. D., Unconditional ideals in Banach spaces, Studia Math. 104 (1993), no. 1, 13–59.

Harmand, P., Werner, D., and Werner, W., $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. https://doi.org/10.1007/BFb0084355

Lima, Å., The metric approximation property, norm-one projections and intersection properties of balls, Israel J. Math. 84 (1993), no. 3, 451–475. https://doi.org/10.1007/BF02760953

Lima, Å., Property $(wM^\ast )$ and the unconditional metric compact approximation property, Studia Math. 113 (1995), no. 3, 249–263. https://doi.org/10.4064/sm-113-3-249-263

Lima, Å. and Oja, E., Ideals of finite rank operators, intersection properties of balls, and the approximation property, Studia Math. 133 (1999), no. 2, 175–186.

Lima, Å. and Oja, E., Ideals of compact operators, J. Aust. Math. Soc. 77 (2004), no. 1, 91–110. https://doi.org/10.1017/S144678870001017X

Lima, Å., Oja, E., Rao, T. S. S. R. K., and Werner, D., Geometry of operator spaces, Michigan Math. J. 41 (1994), no. 3, 473–490. https://doi.org/10.1307/mmj/1029005074

Lima, V. and Lima, Å., Strict u-ideals in Banach spaces, Studia Math. 195 (2009), no. 3, 275–285. https://doi.org/10.4064/sm195-3-6

Nygaard, O. and Põldvere, M., Johnson's projection, Kalton's property $(M^*)$, and $M$-ideals of compact operators, Studia Math. 195 (2009), no. 3, 243–255. https://doi.org/10.4064/sm195-3-4

Oja, E. and Põldvere, M., On subspaces of Banach spaces where every functional has a unique norm-preserving extension, Studia Math. 117 (1996), no. 3, 289–306.

Oja, E. and Põldvere, M., Norm-preserving extensions of functionals and denting points of convex sets, Math. Z. 258 (2008), no. 2, 333–345. https://doi.org/10.1007/s00209-007-0174-8

Phelps, R. R., Uniqueness of Hahn-Banach extensions and unique best approximation, Trans. Amer. Math. Soc. 95 (1960), 238–255. https://doi.org/10.2307/1993289

Põldvere, M., Phelps' uniqueness property for $K(X,Y)$ in $L(X,Y)$, Rocky Mountain J. Math. 36 (2006), no. 5, 1651–1663. https://doi.org/10.1216/rmjm/1181069389

Smith, M. A. and Sullivan, F., Extremely smooth Banach spaces, in “Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976)'', Lecture Notes in Math., vol. 604, Springer, Berlin, 1977, pp. 125--137.
How to Cite
Martsinkevitš, J., & Põldvere, M. (2019). Uniqueness of norm-preserving extensions of functionals on the space of compact operators. MATHEMATICA SCANDINAVICA, 125(1), 67-83. https://doi.org/10.7146/math.scand.a-112071