scholarly journals Weak* fixed point property and the space of affine functions

2020 ◽  
pp. 1
Author(s):  
Emanuele Casini ◽  
Enrico Miglierina ◽  
Łukasz Piasecki
Keyword(s):  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property ◽  
Weak Fixed Point Property ◽  
Affine Functions

scholarly journals A class of spaces with weak normal structure

1994 ◽  
Vol 49 (3) ◽  
pp. 523-528 ◽  
Author(s):  
Brailey Sims
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Point Property ◽  
Weak Fixed Point Property

It has recently been shown that a Banach space enjoys the weak fixed point property if it is ε0-inquadrate for some ε0 < 2 and has WORTH; that is, if then, ║xn — x║ — ║xn + x║ → 0, for all x. We establish the stronger conclusion of weak normal structure under the substantially weaker assumption that the space has WORTH and is ‘ε0-inquadrate in every direction’ for some ε0 < 2.

scholarly journals On \(\ell_1\)-preduals distant by 1

2018 ◽  
Vol 72 (2) ◽  
pp. 41
Author(s):  
Łukasz Piasecki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Wide Class ◽  
Fixed Point Theory ◽  
General Setting ◽  
Point Theory ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property ◽  
Weak Fixed Point Property

For every predual \(X\) of \(\ell_1\) such that the standard basis in \(\ell_1\) is weak\(^*\) convergent, we give explicit models of all Banach spaces \(Y\) for which the Banach-Mazur distance \(d(X,Y)=1\). As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space \(\ell_1\), with a predual \(X\) as above, has the stable weak\(^*\) fixed point property if and only if it has almost stable weak\(^*\) fixed point property, i.e. the dual \(Y^*\) of every Banach space \(Y\) has the weak\(^*\) fixed point property (briefly, \(\sigma(Y^*,Y)\)-FPP) whenever \(d(X,Y)=1\). Then, we construct a predual \(X\) of \(\ell_1\) for which \(\ell_1\) lacks the stable \(\sigma(\ell_1,X)\)-FPP but it has almost stable \(\sigma(\ell_1,X)\)-FPP, which in turn is a strictly stronger property than the \(\sigma(\ell_1,X)\)-FPP. Finally, in the general setting of preduals of \(\ell_1\), we give a sufficient condition for almost stable weak\(^*\) fixed point property in \(\ell_1\) and we prove that for a wide class of spaces this condition is also necessary.

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