Positivity 13 (2009), 427–433
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020427-7, published online August 9, 2008
On the ideal centre of the space of vector
valued integrable functions
Bahri Turan and C¨uneyt C¸evik
Abstract. Let E be a Banach lattice and L
(µ, E) be the space of E-valued
Bochner integrable functions. Some order properties of L
is shown that L
(µ, Z(E)) is the ideal centre of L
(µ, E) and it is obtained
a Radon-Nikodym type theorem for B -integrable functions.
Mathematics Subject Classiﬁcation (2000). 47B65, 46A40, 47B60.
Keywords. Bochner integral, Banach lattice, ideal centre.
Let (Ω, Σ,μ) be a ﬁnite measure space, E be a Banach space, and L
Banach space of E-valued B-integrable functions. The two basic questions here are
what properties of L
(μ, E) are inherited from L
(μ)andE and what properties
of E and L
(μ) are consequences of properties of L
(μ, E). There are many studies
on the subject .
If E is a Banach lattice, then the space L
(μ, E) is also a Banach lattice.
Cartwright  has shown that if E has weakly compact order intervals so does
(μ, E) and consequently L
(μ, E) is Dedekind complete. He also obtained a
The ideal centre plays a fundamental role in the theory of ordered vector
spaces. For example, it can be applied to obtain Radon-Nikodym type theorems
for measures or positive operators [12, Section 145], [1, Chapter 2, Section 8].
Here, we give a characterization for the ideal centre of L
(μ, E) and by using this,
a Radon-Nikodym type theorem is obtained.
2. Preliminaries and Some Order Properties of L
Let (Ω, Σ,μ) be a ﬁnite measure space and E be a Banach space. A function
f :Ω→ E is called simple if there exist x
∈ E and A