On constructing Banach spaces with no unconditional basis

Abstract

Abstract.There exists a weak Hilbert space with no unconditional basis.The aim of this note is to show a rather general construction of Banach spaces with no unconditional basis. As a corollary, one obtains a weak Hilbert space with no unconditional basis, thus answering a question raised some years ago by several authors (cf., e.g., [CS, P]). Our approach is based on techniques first introduced by Johnson, Lindenstrauss, and Schechtman in [JLS] for the study of the Kalton-Peck space [KP]. These techniques were refined b> Ketonen [K] and generalized further by Borzyszkowski [B]. The novelty of the present general approach consists of the use of a simple interpolation trick, which is nevertheless strong enough to allow for the construction of a weak Hilbert space which contains a subspace with no unconditional basis. The main constructionThe standard notation from the Banach space theory used throughout this paper can be found, for example, in [P, TJ].Let us recall that, if (Zfc)£L, is a family of finite-dimensional subspaces of a Banach space X, then the unconditional constant of (Zk)kx=i , denoted by unc(Zk)kxLl , is the infimum of numbers c > 0 such that, for all finite sequences of vectors (xk)k , with xk £ Zk , and for all choices of signs (ek)k , the following holds:If unc(Zk)kx'=l is finite, we call (Zk)k*Ll an unconditional decomposition. A basis (e,)^, in a Banach space X is 1-conditional if unc(span{e,}g[) = 1. If A is a set of positive integers, we denote span({e,}/6/1) in X by X\A .Throughout this paper we fix an interpolation functor which to any 0 < 6 < 1 and any interpolation couple of Banach spaces (Xo, X\), with norms || • ||o and || • ||! respectively, corresponds the space (X0, X\)e , with the norm || • \\e such that:

Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces

“…This provides us with a simple criterion to check whether a given space with a uniform UFDD has l.u.st. : compare the earlier criteria used by Ketonen [10], Borzyszkowski [2] and Komorowski [11]. Using this criterion we establish a general result on the failure of l.u.st.…”

A Weak Hilbert Space That Is a Twisted Hilbert Space

“…Let us recall that the first example of weak Hilbert that is 2 -saturated was given by Edgington [16], while the first example with no unconditional basis was introduced by Komorowski [22]. Later, more examples were given by Komorowski and Tomczak-Jaegermann [23]; see also [2].…”