Positivity 3: 357–364, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
Commutativity of the Arens product in lattice
Department of Mathematics and Applied Mathematics, Potchefstroom University for CHE,
Potchefstroom 2520, South Africa.
(Received: 17 March 1998; accepted: 15 November 1998)
Abstract. Let A be an Abelian Archimedean lattice ordered algebra. The order bidual A
with the Arens product is again a lattice ordered algebra. We show that the order continuous order
is Abelian. This solves an open problem and improves a result of Scheffold, who
proved it for the case of normed lattice ordered algebras. The proof is based on the ‘up-down-up’
approximation of positive elements in the order continuous order bidual (A
by elements in the
A of A in (A
. Components of positive elements in
A are characterized and the
result is applied to the Arens product of f -and almost f -algebras.
Mathematics Subject Classiﬁcation (1991): 46A40, 46A20, 06F25, 13J25.
Key words: Arens product, -algebra, Riesz algebra.
We shall assume throughout that all vector lattices under consideration are Ar-
chimedean. A (real) vector lattice (or Riesz space) which is simultaneously an
associative algebra with the property that xy 0forall0 x, y ∈ A (equiv-
alently, |xy| |x|·|y| for all x, y ∈ A) is called a lattice ordered algebra (also
a Riesz algebra or an -algebra). The vector lattice of all order bounded linear
functionals on A is called the order dual of A and is denoted by A
. Its order dual
is called the order bidual of A and is denoted by A
. The band of all order bounded,
order continuous linear functionals on A
is denoted by (A
and is called the order
continuous order bidual of A. We deﬁne for every x ∈ A an element x ∈ (A
by putting x(f ) := f(x)for all f ∈ A
. The map x → x of A into (A
is a lattice homomorphism and the ideal generated in A
A is order dense in
(see [12, Proposition 1.4.15]). This means that for every positive F ∈ (A
there exists an upwards directed net F
↑ F, with each F
an element of the ideal
a multiplication, called the Arens multiplication (see  and ) is deﬁned
as follows: For all x, y ∈ A, f ∈ A
and F,G ∈ A
we deﬁne f ·x ∈ A
,G·f ∈ A