Abstract. In this paper we study the w * -fixed point property for nonexpansive mappings. First we show that the dual space X * lacks the w * -fixed point property whenever X contains an isometric copy of the space c. Then, the main result of our paper provides several characterizations of weak-star topologies that fail the fixed point property for nonexpansive mappings in ℓ 1 space. This result allows us to obtain a characterization of all separable Lindenstrauss spaces X inducing the failure of w * -fixed point property in X * .

Abstract. The main aim of the present paper is to investigate various structural properties of hyperplanes of c, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and we prove that an hyperplane of c is isometric to the whole space if and only if it is 1-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to ℓ 1 and we give a complete description of the preduals of ℓ 1 under the assumption that the standard basis of ℓ 1 is weak * -convergent.

Abstract. A recent example by the authors (see [3]) shows that an old result of Zippin about the existence of an isometric copy of c in a separable Lindenstrauss space is incorrect. The same example proves that some characterizations of polyhedral Lindenstrauss spaces, based on the result of Zippin, are false. The main result of the present paper provides a new characterization of polyhedrality for the preduals of ℓ 1 and gives a correct proof for one of the older. Indeed, we prove that for a space X such that X * = ℓ 1 the following properties are equivalent:(1) X is a polyhedral space; (2) X does not contain an isometric copy of c;. By known theory, from our result follows that a generic Lindenstrauss space is polyhedral if and only if it does not contain an isometric copy of c. Moreover, a correct version of the result of Zippin is derived as a corollary of the main result.

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The aim of this paper is to study the w * -fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of w * -closed subsets of the dual sphere is equivalent to the w * -fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable w * -fixed point property. The last geometrical notion was introduced by Fonf and Veselý as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be related to a polyhedral concept for the predual space. Indeed, we give a hierarchical structure among various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we obtain an improvement of an old result about the norm-preserving compact extension of compact operators.2010 Mathematics Subject Classification. 47H10, 46B45, 46B25. Key words and phrases. w * -fixed point property, stability of the w * -fixed point property, Lindenstrauss spaces, Polyhedral spaces, ℓ 1 space, Extension of compact operators.