Preserved and unpreserved extreme points A. J. Guirao 1 , V. - PowerPoint PPT Presentation

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Preserved and unpreserved extreme points A. J. Guirao 1 , V. Montesinos 1 , V. Zizler 2 1 Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Spain 2 Alberta University, Edmonton, Alberta, Canada Aleksander Pełczy´ nski Memorial Conference A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Alexander Pełczy´ nski A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Preserved and unpreserved extreme points A. J. Guirao 1 , V. Montesinos 1 , V. Zizler 2 1 Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Spain 2 Alberta University, Edmonton, Alberta, Canada Aleksander Pełczy´ nski Memorial Conference A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Outline Basic facta 1 Earlier and related results 2 Some other definitions and auxiliary results 3 The main results 4 A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Extreme points X Banach. B X closed unit ball, S X unit sphere A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Extreme points X Banach. B X closed unit ball, S X unit sphere x ∈ S X is extreme if x = y + z and y , z ∈ B X ⇒ y = z 2 A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Extreme points X Banach. B X closed unit ball, S X unit sphere x ∈ S X is extreme if x = y + z and y , z ∈ B X ⇒ y = z 2 extreme B X 0 nonextreme A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Extreme points X Banach. B X closed unit ball, S X unit sphere x ∈ S X is extreme if x = y + z and y , z ∈ B X ⇒ y = z 2 extreme X ∗∗ B X ∗∗ B X B X 0 X nonextreme A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Remark If X nonreflexive then B X ∗∗ has extreme points non in X . A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Remark If X nonreflexive then B X ∗∗ has extreme points non in X . X ∗∗ B X ∗∗ B X X A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Remark If X nonreflexive then B X ∗∗ has extreme points non in X . X ∗∗ B X ∗∗ B X X Reason: James’ Theorem A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results preserved extreme points x ∈ S X is preserved extreme whenever extreme of B X ∗∗ A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results preserved extreme points x ∈ S X is preserved extreme whenever extreme of B X ∗∗ B X ∗∗ X all x ∈ S X preserved A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results preserved extreme points Otherwise, unpreserved x ∈ S X is preserved extreme whenever extreme of B X ∗∗ B X ∗∗ B X ∗∗ X X all x ∈ S X unpreserved all x ∈ S X preserved A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results All extreme points are preserved in C ( K ) , L p , 1 ≤ p ≤ ∞ A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results All extreme points are preserved in C ( K ) , L p , 1 ≤ p ≤ ∞ B X ∗∗ X A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results All extreme points are preserved in Question (Phelps’61) C ( K ) , L p , 1 ≤ p ≤ ∞ ∃ unpreserved? B X ∗∗ X A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results All extreme points are preserved in Question (Phelps’61) C ( K ) , L p , 1 ≤ p ≤ ∞ ∃ unpreserved? B X ∗∗ Answer (Katznelson’61) Disk X algebra A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Lemma (Rosenthal) e ∈ S X preserved extreme ⇔ { slices ∋ e } base of w-neighborhoods. e B X A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results B X ∗∗ x 0 B X x 0 + x n 2 B X X x n all S X preserved extreme LUR A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Strongly exposed x 0 A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Denting Strongly exposed x 0 x 0 A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Denting Strongly exposed x 0 x 0 LUR ⇒ strongly exposed A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Denting Strongly exposed x 0 x 0 LUR ⇒ strongly exposed ⇒ denting A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Denting Strongly exposed x 0 x 0 LUR ⇒ strongly exposed ⇒ denting ( w -) Strongly extreme x 0 A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Denting Strongly exposed x 0 x 0 LUR ⇒ strongly exposed ⇒ denting ( w -) Strongly extreme Extreme x 0 x 0 A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Denting Strongly exposed x 0 x 0 LUR ⇒ strongly exposed ⇒ denting ( w -) Strongly extreme Extreme denting ⇒ strongly x 0 x 0 extreme A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Denting Strongly exposed x 0 x 0 LUR ⇒ strongly exposed ⇒ denting ( w -) Strongly extreme Extreme denting ⇒ strongly x 0 x 0 extreme ⇒ w -strongly extreme A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Denting Strongly exposed x 0 x 0 LUR ⇒ strongly exposed ⇒ denting ( w -) Strongly extreme Extreme denting ⇒ strongly x 0 x 0 extreme ⇒ w -strongly extreme (=preserved extreme [Godun–Lin– Troyanski’92]) A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points

Basic facta Earlier and related results Some other definitions and auxiliary results The main results Denting Strongly exposed x 0 x 0 LUR ⇒ strongly exposed ⇒ denting ( w -) Strongly extreme Extreme denting ⇒ strongly x 0 x 0 extreme ⇒ w -strongly extreme (=preserved extreme [Godun–Lin– Troyanski’92]) ⇒ extreme A. J. Guirao, V. Montesinos, and V. Zizler Preserved and unpreserved extreme points