80. Introduction and notation
The well-known notions of the approximation property ( A P ) and the )ounded approximation property ( B A P ) introduced by A. GROTHENDIECK in his famous work [4] play a very important role in the theory of operators in BANACH spaces.
If we know that given spaces have the A P we are able, for example, to describe dual spaces t o many BAXACH spaces of operators (see, e.g., 171, and of course 141); if X has the BAP then we can identify t h e space of nuclear operators in X (up to isomorphism) with a subspace of all integral operators e.t.c. To be short this facts and other ones simplify proofs of many theorems in the geometry of BANACH spaces and in the theory of operator ideals (see [7]). We shdl not he concerned with questions of this type, generally ; mainly, we will consider connections between different approximation conditions in BANACH spaces both of GBOTHENDIECK and more general. Moreover. our main theme in this note will he counterexamples (old and new) to some natural problems in this topics, After ENFLO counterexample to the approximation problem (see 131) the questions of comparing different notions in this dirertion had to herome interesting. , -Z first negative result concerning this was obtained by T. FIGmLand W. B. JOHNSOK in 1211: the A P and the BAP are not equivalent. Tt is this result which lead the author to consider close problem and to investigate in detail questions around the TJIGIEL-JOHNSOK theorem. The present paper is, mainly, a survey of some results obtained in the last 2-3 years. We shall be goncerned only with a part of them.
To generalize the notion of the A P (the B A P ) we have many ways; for instance. one can define the AP, and the BAP, (approximation properties of order q) for q5-0 (so that A P , = A P : see 181, [9], [lU]; for the erne Q > I the definition in
[19] differs from the one in [8] and 191 only in the sense that our AP, is the APq, of SAPHAR) .
We will consider here only the cases when q 5 1. As to case q =-1, the reader is 1) Supported in part by the Swedish Institute Bilateral Scholarship Program