Positivity 11 (2007), 563–574
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040563-12, published online September 26, 2007
On Finite Elements in Lattices of Regular
Zi Li Chen
and Martin R. Weber
Abstract. Let E and F be vector lattices and L
(E, F) the ordered space of
all regular operators, which turns out to be a (Dedekind complete) vector
lattice if F is Dedekind complete. We show that every lattice isomorphism
from E onto F is a ﬁnite element in L
(E, F), and that if E is an AL-space
and F is a Dedekind complete AM-space with an order unit, then each regu-
lar operator is a ﬁnite element in L
(E, F). We also investigate the ﬁniteness
of ﬁnite rank operators in Banach lattices. In particular, we give necessary
and sufﬁcient conditions for rank one operators to be ﬁnite elements in the
vector lattice L
Mathematics Subject Classiﬁcation (2000). 46B42, 47B07, 47B65.
Keywords. Banach lattice, Dedekind complete, Finite element, Regular
operator, Modulus of an operator, Finite rank operator.
In vector lattices of continuous functions on a locally compact Hausdorﬀ space
the ﬁnite functions, i. e. continuous functions with compact support, play a nat-
ural and important role. The notion of a ﬁnite element (see Definition 1) in an
Archimedean vector lattice was introduced in  as some abstract analogue of
continuous functions with compact support.
An Archimedean vector lattice, possessing a sufﬁcient number of ﬁnite ele-
ments, allows (under some additional conditions) a representation as a vector
lattice of (everywhere ﬁnite-valued) continuous functions on a locally compact
σ-compact space such that all ﬁnite elements are represented as ﬁnite functions.
For the details see  and .
A half year stay at the Technical University of Dresden was supported by China Scholarship