This is an announcement for the paper “Unbounded Disjointness Preserving Linear Functionals and Operators” by Anton R Schephttp://arxiv.org/find/math/1/au:+Schep_A/0/1/0/all/0/1.

Abstract: Let $E$ and $F$ be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice $E$, which shows that in this case the unbounded disjointness operators from $E\rightarrow F$ separate the points of $E$. Then we show that every disjointness preserving operator $T: E\rightarrow F$ is norm bounded on an order dense ideal. In case $E$ has order continuous norm, this implies that that every unbounded disjointness preserving map $T: E\rightarrow F$ has a unique decomposition $T=R+S$, where $R$ is a bounded disjointness preserving operator and $S$ is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that $E=C(X)$ with $X$ a compact Hausdorff space, we show that every disjointness preserving operator $T: C(X)\rightarrow F$ is norm bounded on an norm dense sublattice algebra of $C(X)$, which leads then to a decomposition of $T$ into a bounded disjointness operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals.

The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1607.01423