Norm-to-weak upper semi-continuity of the duality and pre-duality mappings
For an element \(x\) in the unit sphere of a Banach space \(X\), the duality mapping is the (set-valued function) given by \(J_{X^\ast}(x)= \{ x^\ast \in X^\ast : 1= \| x^\ast \| = x^\ast (x) \}\). In the case of a dual space, the preduality mapping at an element \(x^\ast \in X^\ast \) is just \(J_{X^{\ast \ast}} \cap X\). The authors use the concept of norm to weak upper semi-continuity for the duality mapping as it appears in [\textit{J. R. Giles}, \textit{D. A. Gregory} and \textit{B. Sims}, Pac. J. Math. 79, 99--109 (1979; Zbl 0399.46012)]. The present paper provides some characterizations of the norm to weak upper semicontinuity of the preduality mapping of a Banach space. For instance, the authors obtain the following result: if the duality mapping is n-w upper-semicontinuous at a point \(x^\ast\) on the unit sphere of a dual Banach space, then \(x^\ast\) attains its norm and the preduality mapping has the same kind of semicontinuity at \(x^\ast\) (Theorem 2.3). A point \(x \) in the unit sphere of \(X\) is said to be very smooth if there is just one element \(\varphi \in X^{\ast \ast \ast }\) satisfying \(\| \varphi \| = 1 = \varphi (x)\). The authors exhibit several assumptions of different nature for which very smoothness goes up from the space to the bidual. For instance, this happens if \(X\) is an \(M\)-ideal in its bidual, a fact that was probably known before. Also, there is an assumption of isomorphic nature implying the same result: the quotient \(X^{\ast \ast }\backslash X\) is a Grothendieck space (see Proposition 3.4). As a consequence, the authors show that for spaces isomorphic to \(C^\ast\)-algebras the above situation holds (Corollary 3.7 and comments that follow). In Section 4, it is proved that the previous results related to very smooth points are no longer true in general by giving an example, whose proof is also suggestive. To summarize, this paper gives a new and interesting view of the norm-weak upper semi-continuity of the duality mapping defined on dual spaces. It also contains new results about the behaviour of very smooth points under special assumptions. The results and techniques are original compared with related results concerning the continuity of the duality mapping previously considered by some other authors. The proof of Theorem 2.3, a clever combination of the use of the local reflexivity principle and weak topologies, is quite interesting. The paper is carefully written and the introduction and comments clarify the situation of the results obtained.
scientific article; zbMATH DE number 1880353