EUDML

In the paper the geometric properties of the positive cone and positive part of the unit ball of the space of operator-valued continuous space are discussed. In particular we show that $ext-ray C_{+} (K, ℒ (H)) = {ℝ_{+} 1_{{k_{0}}} 𝐱 \otimes 𝐱 : 𝐱 \in 𝐒 (H), k_{0} is an isolated point of K}$ $ext 𝐁_{+} (C (K, ℒ (H))) = s-ext 𝐁_{+} (C (K, ℒ (H))) = {f \in C (K, ℒ (H) : f (K) \subset ext 𝐁_{+} (ℒ (H))}$ . Moreover we describe exposed, strongly exposed and denting points.

Smoothness in $ℒ (C (X), C (Y))$

Ryszard Grzaślewicz — 1996

Acta Universitatis Carolinae. Mathematica et Physica

Vertices in $ℒ (H)$

Ryszard Grzaślewicz — 1997

Acta Universitatis Carolinae. Mathematica et Physica

Locally nonconical unit balls in Orlicz spaces

Ryszard Grząślewicz; Witold Seredyński — 2007

Commentationes Mathematicae

The aim of this paper is to investigate the local nonconicality of unit ball in Orlicz spaces, endowed with the Luxemburg norm. A closed convex set $Q$ in a locally convex topological Hausdorff space $X$ is called locally nonconical $(L N C)$ , if for every $x, y \in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U \cap Q) + (y - x) / 2 \subset Q$ . The following theorem is established: An Orlicz space $L^{ϕ} (μ)$ has an $L N C$ unit ball if and only if either $L^{ϕ} (μ)$ is finite dimensional or the measure $μ$ is atomic with a positive greatest lower bound and $ϕ$ satisfies...

Stability of positive part of unit ball in Orlicz spaces

Ryszard Grzaślewicz; Witold Seredyński — 2005

Commentationes Mathematicae Universitatis Carolinae

The aim of this paper is to investigate the stability of the positive part of the unit ball in Orlicz spaces, endowed with the Luxemburg norm. The convex set $Q$ in a topological vector space is stable if the midpoint map $Φ : Q \times Q \to Q$ , $Φ (x, y) = (x + y) / 2$ is open with respect to the inherited topology in $Q$ . The main theorem is established: In the Orlicz space $L^{ϕ} (μ)$ the stability of the positive part of the unit ball is equivalent to the stability of the unit ball.

Extreme points of the positive part of the unit ball and strictly monotone norm.

Ryszard Grzaslewicz; Helmut H. Schaefer — 1991

Mathematische Zeitschrift

Denting point in the space of operator-valued continuous maps.

Ryszard Grzaslewicz; Samir B. Hadid — 1996

Revista Matemática de la Universidad Complutense de Madrid

In a former paper we describe the geometric properties of the space of continuous functions with values in the space of operators acting on a Hilbert space. In particular we show that dent B(L(H)) = ext B(L(H)) if dim H < 8 and card K < 8 and dent B(L(H)) = 0 if dim H < 8 or card K = 8, and x-ext C(K,L(H)) = ext C(K,L(H)).