Resumen de 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⊥ is the kernel of a norm-one projection P on the dual space Z∗. If Y is an ideal in Z with respect to a projection on Z∗ 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 K(X,Y) of compact operators between Banach spaces X and Y to the larger space K(X,Z) under the assumption that Y is a strict ideal in Z. Our main results are: (1) if y∗ is an extreme point of BY∗ having a unique norm-preserving extension to Z, and x∗∗∈BX∗∗, then the only norm-preserving extension of the functional x∗∗⊗y∗∈K(X,Y)∗ to K(X,Z) is x∗∗⊗z∗ where z∗∈Z∗ is the only norm-preserving extension of y∗ to Z; (2) if K(X,Y) is an ideal in K(X,Z) and Y has Phelps' property U in its bidual Y∗∗ (i.e., every bounded linear functional on Y admits a unique norm-preserving extension to Y∗∗), then K(X,Y) has property U in K(X,Z) whenever X∗∗ has the Radon-Nikodým property.