A Banach space without a basis which has the bounded approximation property

can be approximated by finite rank operators uniformly on compact sets. It is clear that Xhasabasis ~ XhasBAP =~ XhasAP The fact that the converse implication to the second one does not hold in general was discovered by Figiel and Johnson [8] soon after Enflo's example [7] of a space without AP. The main purpose of this paper is to show that also the implication "BAP=~basis" does not hold in general; this answers problems asked by a number of authors (e.g. [14], [18], [27]). Before stating the result, we recall more notation. For a given basis (x n) of X one denotes by bc (x n) (the basis constant of ix~)) the smallest K such that

[1]  E. Gluskin,et al.  Diameter of the Minkowski compactum is approximately equal to n , 1981 .

[2]  On the existence and uniqueness of complex structure and spaces with “few” operators , 1986 .

[3]  William B. Johnson,et al.  The approximation property does not imply the bounded approximation property , 1973 .

[4]  D. R. Lewis.  Finite dimensional subspaces of $L_{p}$ , 1978 .

[5]  A. Pełczyński.  All separable Banach space admit for every ε>0 fundamental total and bounded by 1 + ε biorthogonal sequences , 1976 .

[6]  William B. Johnson,et al.  On bases, finite dimensional decompositions and weaker structures in Banach spaces , 1971 .

[7]  Nicole Tomczak-Jaegermann,et al.  On nearly euclidean decomposition for some classes of Banach spaces , 1980 .

[8]  A. Pełczyński,et al.  Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basic , 1971 .

[9]  S. Szarek.  The finite dimensional basis problem with an appendix on nets of Grassmann manifolds , 1983 .

[10]  Gilles Pisier.  Counterexamples to a conjecture of Grothendieck , 1983 .

[11]  G. Pisier,et al.  Un théorème sur les opérateurs linéaires entre espaces de Banach qui se factorisent par un espace de Hilbert , 1980 .

[12]  I. Singer,et al.  Bases in Banach Spaces II , 1970 .

[13]  Real isomorphic complex Banach spaces need not be complex isomorphic , 1986 .

[14]  Per Enflo,et al.  A counterexample to the approximation problem in Banach spaces , 1973 .

[15]  S V Bočkarev,et al.  EXISTENCE OF A BASIS IN THE SPACE OF FUNCTIONS ANALYTIC IN THE DISK, AND SOME PROPERTIES OF FRANKLIN'S SYSTEM , 1974 .

[16]  A superreflexive Banach space which does not admit complex structure , 1986 .

[17]  J. Lindenstrauss,et al.  Absolutely summing operators in Lp spaces and their applications , 1968 .

[18]  A. Grothendieck,et al.  Produits Tensoriels Topologiques Et Espaces Nucleaires , 1966 .

[19]  Large subspaces ofl∞n and estimates of the gordon-lewis constantand estimates of the gordon-lewis constant , 1980 .

[20]  W. Johnson.  Factoring compact operators , 1971 .

[21]  J. Lindenstrauss,et al.  The ℒp spaces , 1969 .

[22]  A. Szankowski.  Subspaces without the approximation property , 1978 .