On positive operator-valued continuous maps

Ryszard Grzaślewicz

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 3, page 499-505
  • ISSN: 0010-2628

Abstract

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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 ⊗ 𝐱 : 𝐱 ∈ 𝐒 ( H ) , k 0 is an isolated point of K } ext 𝐁 + ( C ( K , ℒ ( H ) ) ) = s-ext 𝐁 + ( C ( K , ℒ ( H ) ) ) = { f ∈ C ( K , ℒ ( H ) : f ( K ) ⊂ ext 𝐁 + ( ℒ ( H ) ) } . Moreover we describe exposed, strongly exposed and denting points.

How to cite

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Grzaślewicz, Ryszard. "On positive operator-valued continuous maps." Commentationes Mathematicae Universitatis Carolinae 37.3 (1996): 499-505. <http://eudml.org/doc/247921>.

@article{Grzaślewicz1996,
abstract = {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 $\text\{ext-ray\} \text\{C\}_+(K,\mathcal \{L\}(H)) = \lbrace \mathbb \{R\}_+ \{\mathbf \{1\}\}_\{\lbrace k_0\rbrace \} \mathbf \{x\}\otimes \mathbf \{x\} : \mathbf \{x\}\in \mathbf \{S\}(H), k_0 \text\{ is an isolated point of \} K\rbrace $$\text\{ext\} \mathbf \{B\}_+(\text\{C\}(K,\mathcal \{L\}(H))) = \text\{s-ext \} \mathbf \{B\}_+(\text\{C\}(K,\mathcal \{L\}(H)))=\lbrace f\in \text\{C\}(K,\mathcal \{L\}(H) : f(K)\subset \text\{ext \} \mathbf \{B\}_+(\mathcal \{L\}(H))\rbrace $. Moreover we describe exposed, strongly exposed and denting points.},
author = {Grzaślewicz, Ryszard},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {exposed point; denting point; Hilbert space; positive operator; exposed point; denting point; Hilbert space; positive operator},
language = {eng},
number = {3},
pages = {499-505},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On positive operator-valued continuous maps},
url = {http://eudml.org/doc/247921},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Grzaślewicz, Ryszard
TI - On positive operator-valued continuous maps
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 3
SP - 499
EP - 505
AB - 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 $\text{ext-ray} \text{C}_+(K,\mathcal {L}(H)) = \lbrace \mathbb {R}_+ {\mathbf {1}}_{\lbrace k_0\rbrace } \mathbf {x}\otimes \mathbf {x} : \mathbf {x}\in \mathbf {S}(H), k_0 \text{ is an isolated point of } K\rbrace $$\text{ext} \mathbf {B}_+(\text{C}(K,\mathcal {L}(H))) = \text{s-ext } \mathbf {B}_+(\text{C}(K,\mathcal {L}(H)))=\lbrace f\in \text{C}(K,\mathcal {L}(H) : f(K)\subset \text{ext } \mathbf {B}_+(\mathcal {L}(H))\rbrace $. Moreover we describe exposed, strongly exposed and denting points.
LA - eng
KW - exposed point; denting point; Hilbert space; positive operator; exposed point; denting point; Hilbert space; positive operator
UR - http://eudml.org/doc/247921
ER -

References

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