Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinitedimensionality, separability, and closeness are all indispensable.

Two existence results for monotone trajectories of diierential inclusions x 0 (t) 2 F ? t; x(t) in a separable Banach space are obtained; they extend in two directions previous ones due to Aubin-Cellina, Deimling and Haddad.

In terms of fragmentability, we describe a new class of Banach spaces which may be c0-saturated but do not contain weak-Gδ open bounded subsets. In particular, none of these spaces is isomorphic to a separable polyhedral space.

We obtain necessary and sufficient conditions in the Large Deviation Principle for random upper semicontinuous functions on a separable Banach space. The main tool is the recent work of Arcones on the LDP for empirical processes.

We study the comparison between the Aumann and the Bochner integrals for integrably non empty bounded closed convex valued multifunctions in a separable Banach space when it is not possible to apply embedding theorems.

Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C∞-smooth bump which depends locally on finitely many coordinates.

We show that on a separable Banach space most Lipschitz functions have maximal Clarke subdiierential mappings. In particular, the generic nonexpansive function has the dual unit ball as its Clarke subdiierential at every point. Diverse corollaries are given.

In this note, we provide a starlike and normal tiling in any separable Banach space. That means, there are positive constants r R (not depending on the space) such that every tile of is starlike, contains ball radius contained .

We realize the noncommutative Gurarij space NG defined by Oikhberg as the Fräıssé limit of the class of finite-dimensional 1-exact operator spaces. As a consequence we deduce that the noncommutative Gurarij space is unique up to completely isometric isomorphism, homogeneous, and universal among separable 1-exact operator spaces. Moreover we show that NG is isometrically isomorphic to the Gurari...

We characterize the class of separable Banach spaces X such that for every continuous function f : X → R and for every continuous function ε : X → (0,+∞) there exists a C smooth function g : X → R for which |f(x)− g(x)| ≤ ε(x) and g′(x) 6= 0 for all x ∈ X (that is, g has no critical points), as those infinite dimensional Banach spaces X with separable dual X∗. We also state sufficient condition...