Positivity 1: 381–390, 1997.
© 1997 Kluwer Academic Publishers. Printed in the Netherlands.
The Norm of a Complex Banach Lattice
J.M.A.M. VAN NEERVEN
Department of Mathematics, TU Delft, P.O. Box 5031, 2600 GA Delft, The Netherlands
(Received: 8 September 1997; Accepted: 4 November 1997)
Abstract. In this note we study the problem how the complexiﬁcation of a real Banach space can
be normed in such a way that it becomes a complex Banach space from the point of view of the
theory of cross-norms on tensor products of Banach spaces. In particular we show that the norm of a
complex Banach lattice can be interpretated in terms of the l-tensor product of real Banach lattices.
Mathematics Subject Classiﬁcations (1991): 46B42, 46M05
Key words: Complexiﬁcation of Banach space, cross-norm, Banach lattice, l-tensor product, cone
absolutely summing operator, operator ideal
Let X be a real Banach space and let X
be its complexiﬁcation. There are various
ways to introduce a norm on X
which makes it into a complex Banach space. In
this note we study this problem systematically by means of cross-norms. The main
idea is the following. Regarding X
as the tensor product X ⊗
the realiﬁcation (X
with X ⊗
(both tensor products are with respect
), we show that every ‘reasonable’ norm making X
into a complex Banach
space is induced by a complex-homogenous cross-norm on X⊗
Thus the study of complex norms of X
is reduced to that of cross-norms on
This is applied to Banach lattices as follows. The complexiﬁcation E
of a real
Banach lattice E is a complex Banach lattice in the norm z:=|z|,where|z|
is the complex modulus of an element z ∈ E
, which is deﬁned in Section 2 below.
We show that this norm is induced by the l-norm on E ⊗
. This is the cross-norm
induced on E ⊗
by the operator ideal L
) of cone absolutely summing
It is interesting to observe at this point that there exist complex Banach spaces
which cannot be obtained as the complexiﬁcation of a real Banach space. The ex-
istence of such a space was proved by Bourgain  using probabilistic arguments;
the ﬁrst explicit example was constructed by Kalton .
This work was carried out while the author was afﬁliated at the Centre for Mathematics and
Computer Science (CWI) in Amsterdam.
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