Positivity 11 (2007), 119–121
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010119-3, published online October 13, 2006
Solutions of Two Problems in the Theory
of Disjointness Preserving Operators
Abstract. In this note, our aim is to solve two problems in the theory of
disjointnesss preserving operators which are posed by Abramovich and Kitover
using some examples constructed by the same authors.
Mathematics Subject classiﬁcation. 47B60.
Keywords. Disjointness preserving operator, band operator, rich center.
Recall that a (linear) operator T : E → F between vector lattices is disjoint-
ness preserving if T sends elements disjoint in E to elements disjoint in F .
The reader is referred to  for the precise statements and the history regard-
ing the operators preserving disjointness. Throughout the work all vector lattices
are assumed to be Archimedean. For the basic theory on vector lattices and for
unexplained terminology we refer to , , .
Deﬁnition 1. Let T : E → F be an operator. T is called a band operator if T (B)
is a band in F for each band B in E.
We refer to  for the properties of ideal operators which have similar deﬁ-
nition. The properties of band operators may be a topic of another paper, so it
is not considered here. Let T be a bijective disjointness preserving operator. It is
well known that T is a band operator whenever either E is Dedekind complete
vector lattice or T
preserves disjointness, in other words T is a d-isomorphism
. Note that, If T is a d-isomorphism then T and T
are band operators.
Recall that an operator π : E → E on a vector lattice is said to be band
preserving whenever π(B) ⊆ B holds for each band B of E. An orthomorphism
is a band preserving operator that is also order bounded. The collection of all or-
thomorphisms on a vector lattice E will be denoted Orth(E). The ideal generated
by the identity operator I in Orth(E) is called the ideal center of E and will be
denoted by Z(E).
Proposition 2. Let T : E → F be a bijective operator. Then the following state-