Positivity 7: 47–59, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Bounded Variation and Tensor Products of Banach
and ARNOUD VAN ROOIJ
Department of Mathematics, University of Mississippi, University, MS 38677, USA.
Catholic University, Department of Mathematics, Toernooiveld, 6525 ED Nijmegen, The
Abstract. We introduce bilinear maps of order bounded variation, semivariation and norm bounded
variation. We use these notions to extend the knowledge of the projective tensor product of Banach
Tensor products have been an important staple in the general theory of Banach
spaces ever since Schatten’s paper  and, of course, Grothendieck’s memoir .
A study of tensor products in the theory of Banach lattices arose much later. In
fact, the inception of a general tensor product for Archimedean Riesz spaces by
Fremlin dates back only to 1972, followed by a theory of tensor products for
Banach lattices in 1974. The amount of information about tensor products of Riesz
spaces still appears rather limited compared to the explosive growth of that subject
in the theory of Banach spaces. One reason for this lack of information lies in the
difﬁculty of the subject matter. Seemingly innocent and natural questions turn out
to be surprisingly hard and the very construction of tensor products of Riesz spaces
raises questions about what should be their deﬁning universal property. It is, in part,
the latter question that we address here.
In  Fremlin deﬁned the Archimedean Riesz space tensor product E
the Archimedean Riesz spaces E and F. His deﬁnition required that for every
Archimedean Riesz space G and for every Riesz bimorphism φ : E×F → G there
exists a unique Riesz homomorphism φ
⊗F → G such that φ
(x ⊗ y) =
φ(x,y) for all x ∈ E, y ∈ F. As Fremlin himself remarked, the existence of
this Archimedean tensor product is less remarkable than the additional universal
property that it possesses: For every uniformly complete Riesz space G and every
bipositive map T : E × F → G there exists a unique positive map T
G such that T
(x ⊗ y) = T (x, y) for all x ∈ E, y ∈ F. Is there a similar
property for bilinear maps that are not necessarily bipositive? A theory of tensor
products should link the bilinear maps that appear in the universal property with
the natural space of linear mappings on the tensor product. For instance, in case of