Uniqueness of norm-preserving extensions of functionals on the space of compact operators - Martsinkevitš, Julia; Põldvere, Märt (MATHEMATICA SCANDINAVICA, 2019)
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 B_Y^* having a unique norm-preserving extension to Z, and x^** B_X^** , 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.