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Positive approximation properties of Banach lattices. (English) Zbl 1411.46016

Recall that a Banach space \(X\) has the approximation property (AP) if, for every compact subset \(K\) of \(X\) and every \(\varepsilon >0\), there exists a finite rank operator \(S\) on \(X\) such that \(\|Sx-x\|<\varepsilon\) for all \(x \in K\). If, in addition, the operator \(S\) can be chosen with norm less than or equal to some \(\lambda \geq 1\), it is said that the space has the bounded approximation property (BAP). Also, in the case when \(X\) is Banach lattice, \(X\) has the positive approximation property or the bounded positive approximation property when the operator \(S\) is positive.
On the other hand, if the dual of a Banach space has the BAP, then the underlying space has the BAP and the converse is not true in general. However, W. B. Johnson and T. Oikhberg showed in [Ill. J. Math. 45, No. 1, 123–137 (2001; Zbl 1004.46008)] that, if a Banach space has the extendably locally reflexive property (this property was introduced by T. Oikhberg and H. P. Rosenthal [J. Funct. Anal. 179, No. 2, 251–308 (2001; Zbl 1064.47072)]), then a Banach space has the BAP if and only if its dual has it.
In this article, this last result is obtained in the case of the bounded positive approximation property. For this, the author proves an equivalent reformulation of the extendably locally reflexive property and uses this reformulation to introduce a positive version of this notion, the positively extendably locally reflexive property. Then he shows that, for a Banach lattice \(X\), the dual space of \(X\) has the bounded positive approximation property if and only if \(X\) has the positive locally reflexive property and the bounded positive approximation property.
Also, some other reformulations of the approximation property, obtained in [C. Choi et al., Can. Math. Bull. 52, No. 1, 28–38 (2009; Zbl 1187.46012)] and [J. M. Kim, J. Math. Anal. Appl. 345, No. 2, 889–891 (2008; Zbl 1154.46008)] are obtained for the positive approximation property.

MSC:

46B28 Spaces of operators; tensor products; approximation properties
46B07 Local theory of Banach spaces
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References:

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