Reflexivity of cyclic Banach spaces

Abstract

In a Banach space l with an unconditional basis { en } the projections E(a); o-CN= {1, 2, 3, * **, n, ... } defined by E(o)(Zn=1anen) =Ena anen; =x ianen(Ex form a u-complete atomic Boolean algebra of projections & for which there exists a vector x0o?E (for instance, x0 = EZ0en/2nIlenIl) such that Y =clm {Exo I EE}. Viewed from this point, the Banach spaces having unconditional basis form a subclass of the family of cyclic spaces = clm { Pxo I P E 63 } for some xOCG and a u-complete (not-necessarily atomic) Boolean algebra of projections 63 on X. Cyclic spaces have been introduced by W. G. Bade [1], [2] in connection with the multiplicity theory for spectral operators on Banach spaces. A typical example is L1(0, 1), the space of all integrable functions on [0, 1], which has no unconditional basis (cf. A. Pelczynski [13, Proposition 9]) but is a cyclic space with respect to the Boolean algebra of projections consisting of multiplications by characteristic functions. W. G. Bade suggested recently in a discussion that it might follow from the theory of normed lattices that a cyclic space is reflexive provided its second conjugate is separable. Using a theorem of T. Ogasawara [12] on normed Riesz spaces we shall be able to prove in the present note that reflexivity of a cyclic space X is insured by the condition (weaker than separability of the second conjugate) that neither 11 nor c0 would be isomorphic to a subspace of X. This result generalizes a well-known characterization of reflexivity for spaces with unconditional bases given by R. C. James [5]. Other properties of Banach spaces in connection with Boolean algebras of projections have been described recently in [6], [7], [10], [14].