Positivity 8: 423–441, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Unconditional Decompositions in Subspaces of
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada.
Received and accepted 10 March 2003
Abstract. If a Banach space X is not isomorphic to a Hilbert space then l
X contains a subspace
which has a UFDD, but does not admit a UFDD with a uniform bound for the dimensions of the
It is well-known that even if a Banach space has an unconditional ﬁnite-dimensional
Schauder decomposition (UFDD) into two-dimensional subspaces, it may fail to
have an unconditional basis. The ﬁrst example of such phenomenon was the Kalton–
Peck space . In fact, this space does not even have local unconditional structure,
as it was proved by Johnson et al. . Their technique was later reﬁned by Ketonen
 and Borzyskowski , who used it for subspaces of L
, and subsequently
generalized in the work of Komorowski and Tomczak-Jaegermann [7–10], where a
general method of constructing subspaces without unconditional basis (which still
admit a two-dimensional UFDD) was developed.
In this paper we continue the investigation of properties related to uncondition-
ality in Banach spaces which admit a UFDD.
We obtain the following characterization of a Hilbert space: a Banach space X
is isomorphic to a Hilbert space if and only if for every subspace Y of l
exists k 1 such that Y admits a k-dimensional UFDD.
The main step consists of constructing, for a non-hilbertian Banach space X
with ﬁnite cotype and for all integers k 2, a subspace of l
X which has
a k-dimensional UFDD and for which k is minimal with such property. Similar
constructions were previously done for subspaces of L
,1p<2 (in 1).
Another consequence of this construction, as it was pointed out to the author by
Valentin Ferenczi, is that l
X contains at least countably many mutually non-
isomorphic inﬁnite dimensional subspaces, when X is a non-hilbertian Banach
space with ﬁnite cotype.