UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND ℓ∞

Abstract

AbstractIn this paper we study non-complemented spaces of operators and the embeddability of ℓ∞in the spaces of operatorsL(X,Y),K(X,Y) andKw*(X*,Y). Results of Bator and Lewis [2,3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416;Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99(1991), 125–130;Math. Proc. Camb. Phil. Soc.111(1992), 331–335;Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25(1982), 78–81) and Kalton [16] (Math. Ann.208(1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spacesW(X,Y) andK(X,Y) in the spaceL(X,Y), as well as the complementation ofK(X,Y) inW(X,Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc.108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.