Positivity 4: 289–291, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
An Extension of Kakutani’s Theorem on Abstract
-spaces to the Case of any Positive p
Mathematical Institute, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The
1. Introduction and Summary
For p ∈[1, ∞), Kakutani’s theorem on abstract L
-spaces (see [1, page 71])
gives a characterization of L
-spaces within the class of Banach lattices. In the
talk we presented an extension of this theorem that includes the case p ∈ (0, 1).
The extension characterizes L
-spaces for p ∈ (0, ∞), within the class of locally
solid Riesz spaces (see [1, page 33]) and has the advantage of not distinguishing
between the case p ∈[1, ∞) (when L
is a Banach lattice) and the case p ∈ (0, 1)
is no norm).
Even for p ∈[1, ∞), the extension turns out to be a generalization, since the
subadditivity of ·
is no longer required.
We ﬁrst recall some deﬁnitions and properties relating to L
Let (S, A,µ)be a measure space and p ∈ (0, ∞).
Let M be the (Riesz) space of all measurable functions f : S → R with
identiﬁcation of functions that are µ-almost everywhere equal and let L
(µ) be the
(Riesz) subspace of M consisting of all f ∈ M for which f
-spaces are order isomorphic. Indeed,
: x →
for x 0
for x 0
deﬁnes an order isomorphism R → R with inverse φ
,sothatf → f
is an order isomorphism M → M. From the equality f
, it follows
that f → f
- spaces are order isomorphic.
Now set E := L
(µ) and N :=·
Then (E, N) has the following properties.