C*-Algebras That AreI-Rings

Banach space characterizations of unitaries: A survey

“…It is worth mentioning that, after Gregory's paper, most authors have preferred the terminology of strong subdifferentiability of the norm instead of that of norm-norm upper semi-continuity of the duality mapping. This happens in particular in the papers where the points of norm-norm upper semi-continuity for the duality mapping of C * -algebras, JB * -triples, and real JB * -triples are determined (see [16], [11], and [10], respectively).…”

Strong ball proximinality of the space of compact operators

“…Using [2, Lemma 2] and polar decomposition, we can see that the norm of B(H) is strongly subdifferentiable at an operator T ∈ B(H) if and only if the norm of B(H) is strongly subdifferentiable at |T | = (T * T ) 1/2 . Now, since the collection of all operators at which the norm of B(H) is strongly subdifferentiable is a dense subset of B(H) (see [2,Theorem 3]), the set of those positive operators at which the norm of B(H) is strongly subdifferentiable is a dense subset of the set of all positive operators on B(H). Hence we have the following results.…”

Subdifferentiability of the norm and the Banach-Stone theorem for real and complex JB*-triples

“…The norm of X is said to be strongly-subdifferentiable at a morn-one point x ∈ X whenever the limit lim α→0 + x + αy − 1 α exists uniformly for y in the closed unit ball of X. The points of strong subdifferentiability for the norm of a C * -algebra were characterized by Contreras, Payá and Werner in [6]. Recently, Becerra-Guerrero and Rodríguez-Palacios have completely described the points of strong subdifferentiability for the norm of a (complex) JB * -triple [3].…”