Positivity 5: 297–321, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Embeddings of Banach Spaces Into Banach Lattices
and the Gordon–Lewis Property
and N.J. NIELSEN
Department of Mathematics, University of Missouri, Columbia MO 65211 USA. E-mail:
Department of Mathematics and Computer Science, Odense
University, Campusvej 55, DK-5230 Odense M, Denmark. E-mail: firstname.lastname@example.org
Abstract. In this paper we ﬁrst show that if X is a Banach space and α is a left invariant crossnorm
⊗ X, then there is a Banach lattice L and an isometric embedding J of X into L,sothat
I ⊗ J becomes an isometry of
J(X).HereI denotes the identity operator on
J(X) the canonical lattice tensor product. This result is originally due to G. Pisier
(unpublished), but our proof is different. We then use this to prove the main results which characterize
the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.
Mathematics Subject Classiﬁcation (2000): 46B40, 46B42
In this paper we investigate embeddings of Banach spaces into Banach lattices
which preserve a certain tensorial structure given a priori. This is then used to char-
acterize the Gordon–Lewis property GL and related structures in Banach spaces.
The basic result, Theorem 1.7, states that if X is a Banach space and α isaleft
tensorial crossnorm on
⊗ X (see Section 0 for the deﬁnition), then there exist a
Banach lattice L and an isometric embedding J of X into L so that I ⊗ J becomes
an isometric embedding of
J(X).HereI denotes the identity
X the canonical lattice tensor product. This result was
originally proved by Pisier  (unpublished), but our construction of the Banach
lattice L is quite different from his. It is a modiﬁcation of a construction given by
the second named author and presented at a conference in Columbia, Missouri in
1994 and is based on our Theorem 1.5 below.
This result is then used to prove that a Banach space X has GL
if and only if
it embeds into a Banach lattice L so that every absolutely summing operator from
X to a Hilbert space extends to an absolutely summing operator deﬁned on L.Ina
similar manner we prove that X has the general GL-propery if and only if it embeds
into a Banach lattice L so that every absolutely summing operator from X to an
arbitrary Banach space Y extends to a cone-summing operator from L to Y .Some
Supported by NSF grant DMS 970618.
Supported in part by the Danish Natural Science Research Council, grants 9503296 and