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Uniform non l -ness of l∞-sums of Banach spaces

Kato M., Tamura T.

Commentationes Mathematicae

2009

Vol. 49, [Z] 2

179-187

We shall characterize the uniform non-l {...} -ness of `l∞-sums of Banach spaces (X1 ⊕ź ź ź⊕Xm)1. As applications, some results on super-reflexivity and the fixed point property for nonexpansive mappings will be presented.

A new geometrical constant of Banach spaces and the uniform normal structure

Mitani K-I., Saito K-S.

Commentationes Mathematicae

2009

Vol. 49, [Z] 1

3-13

We introduce and study a new geometrical constant γXψ of a Banach space X, by using the notion of ψ-direct sum given in [Y. Takahashi, M. Kato and K.-S. Saito, J. Inequal. Appl. 7 (2002), 179-186]. At rst, we characterize uniform non- squareness in terms γXψ Moreover, we consider Banach spaces having uniform normal structure.

Uniform non-ln1-ness of l_1-sums of Banach spaces

Kato M., Tamura T.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2007

Vol. 47, [Z] 2

161-169

We shall characterize the uniform non-`n1-ness of the `1-sum (X1 +ź ź ź + X_m)_1 of a finite number of Banach spaces X_1, ź ź ź ,X_m. Also we shall obtain that (X_1 +ź ź ź +X_m)_1 is uniformly non-lm+1 if and only if all X_1, . . . ,X_m are uniformly non-square (note that (X_1 + ź ź ź +X_m)_1 is not uniformly non-lm1). Several related results will be presented.