On the extension property for compact operators

Abstract

1. Introduction. The main theorem of this note shows the equiv­ alence of several extension properties for operators and contains also some geometrical characterizations of the spaces having these proper­ ties. In particular a characterization of such spaces is given in terms of intersection properties of their cells, similar to that given by Nachbin for (Pi spaces [8], Our theorem extends previous results of Grothendieck [4]. Some applications are given, among them a new character­ ization of C{K) spaces. In this connection some problems raised by Aronszajn and Panitchpakdi [l], Grothendieck [4] and Nachbin [8; 9] are solved. Notations. All Banach spaces are assumed to be over the reals. Sx(xo, r) denotes the cell {x;xGl, ||x — xo Sr}. A Banach space X is called a (Px space if from any Z containing X there is a projection on X with norm ^X (see Day [2, pp. 94-96]). We say that a Banach space has the metric approximation property (M.A.P.) if for every compact subset K of X and every e>0 there is a compact operator from X into itself such that || T = 1 and || Tx-x ue for x£K. A (possibly) stronger property was introduced by Grothendieck [3, pp. 187-191]. He proved that the common Banach spaces have the M.A.P. It is an open problem whether there exists a Banach space which does not have the M.A.P. 2. The main results. We state now the main result of this note (the equivalencies l<-*2<-»3<-»5 are due to Grothendieck [4]). In the extension properties stated below F, Z and V will be arbitrary Ba­ nach spaces satisfying Z~2)Y, TO^and the indicated restrictions (if any).