Remarks on retracting balls on spherical caps in \(c_{0}\), \(c\), \(l^{\infty }\) spaces

For any infinite dimensional Banach space there exists a lipschitzian retraction of the closed unit ball B onto the unit sphere S. Lipschitz constants for such retractions are, in general, only roughly estimated. The paper is illustrative. It contains remarks, illustrations and estimates concerning optimal retractions onto spherical caps for sequence spaces with the uniform norm.

Regularity in Topological Modules

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and m∗∈M∗ is a continuous linear functional on M which is open as a map between topological spaces, then m∗−1(int(B)) is regular open and m∗−1(B) is regular closed, for B any closed-unit zero-neighborhood in R.

Absolutely compatible pairs in a von Neumann algebra

Let a, b be elements in a unital C∗-algebra with 0 ≤ a, b ≤ I. The element a is absolutely compatible with b if |a − b| + |I − a − b| = I. In this note, some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra are found. These characterizations are applied to measure how far are two absolute compatible positive elements in the closed unit ball from being mutually orthogonal or commuting. In the case of 2 by 2 matrices, the results admit a geometric interpretation. Namely, non-commutative matrices of the form a = ( t α ) and b = ( x β ) with x, t ∈ (0, 1)\{ 1 }, |α|2 < t(1 − t) α¯ 1 − t β 1 − x 2 and |β|2 < x(1 − x), are absolutely compatible if, and only if, the corresponding point b = (x, lRe(β), S'm(β)) in R3 lies in the ellipsoid Ea = {x ∈ R3 : d2(x, a) + d2(x, a,) = 1}, where d2 denotes the Euclidean distance in R3, and the elements a and a, are (t, lRe(α), S'm(α)) and (1 − t, −lRe(α), −S'm(α)), respectively. The description of absolutely compatible pairs of positive 2 by 2 matrices is applied to determine absolutely compatible pairs of positive elements in the closed unit ball of Mn.

Geometry of balanced and absorbing subsets of topological modules

We define the concepts of balanced set and absorbing set in modules over topological rings, which coincide with the usual concepts when restricting to topological vector spaces. We show that in a topological module over an absolute semi-valued ring whose invertibles approach [Formula: see text], every neighborhood of [Formula: see text] is absorbing. We also introduce the concept of total closed unit neighborhood of zero (total closed unit) and prove that the only total closed unit of the quaternions [Formula: see text] is its closed unit ball [Formula: see text]. On the other hand, we also prove that if [Formula: see text] is an absolute semi-valued unital real algebra, then its closed unit ball [Formula: see text] is a total closed unit. Finally, we study the geometry of modules via the extreme points and the internal points, showing that no internal point can be an extreme point and that absorbance is equivalent to having [Formula: see text] as an internal point.

$k$-smoothness: an answer to an open problem

The aim of this paper is to characterize the $k$-smooth points of the closed unit ball of $\mathcal{K}(\mathcal{H}_1;\mathcal{H}_2)$. In this paper we also answer a question posed by A. Saleh Hamarsheh in 2015.

Biduality and density in Lipschitz function spaces

For pointed compact metric spaces $(X,d)$, we address the biduality problem as to when the space of Lipschitz functions $\mathrm{Lip}_0 (X,d)$ is isometrically isomorphic to the bidual of the space of little Lipschitz functions $\mathrm{lip}_0 (X,d)$, and show that this is the case whenever the closed unit ball of $\mathrm{lip}_0 (X,d)$ is dense in the closed unit ball of $\mathrm{Lip}_0 (X,d)$ with respect to the topology of pointwise convergence. Then we apply our density criterion to prove in an alternative way the real version of a classical result which asserts that $\mathrm{Lip}_0 (X,d^\alpha )$ is isometrically isomorphic to $\mathrm{lip}_0 (X,d^\alpha )^{**}$ for any $\alpha \in (0,1)$.

Distortion Type Theorems for Functions in the Logarithmic Bloch Space

We establish distortion type theorems for locally schlicht functions and for functions having branch points satisfying a normalized Bloch condition in the closed unit ball of the logarithmic Bloch spaceBlog. As a consequence of our results we have estimations of the schlicht radius for functions in these classes.

Weak sequential convergence in the dual of compact operators between Banach lattices

For several Banach lattices E and F, if K(E,F) denotes the space of all compact operators from E to F, under some conditions on E and F, it is shown that for a closed subspace M of K(E,F), M* has the Schur property if and only if all point evaluations M1(x) = {Tx : T ? M1} and ~M1(y*) = {T* y* : T ? M1} are relatively norm compact, where x ? E, y* ? F* and M1 is the closed unit ball of M.

STRUCTURE TOPOLOGY AND EXTREME OPERATORS

We say that a Banach space $X$ is ‘nice’ if every extreme operator from any Banach space into $X$ is a nice operator (that is, its adjoint preserves extreme points). We prove that if $X$ is a nice almost $CL$-space, then $X$ is isometrically isomorphic to $c_{0}(I)$ for some set $I$. We also show that if $X$ is a nice Banach space whose closed unit ball has the Krein–Milman property, then $X$ is $l_{\infty }^{n}$ for some $n\in \mathbb{N}$. The proof of our results relies on the structure topology.