保无界不交性的线性泛函与算子
Unbounded Disjointness Preserving Linear Functionals and Operators
摘要：
设$E$和$F$是Banach格.首先证明了保不交线性泛函可分离任意无穷维Banach格$E$的点，这表明在这种情况下，来自$E\to F$的无界不交算子可分离$E$的点.然后证明了每一个保持不交性的算子$T:E\to F$在一个序稠密理想上是范数有界的。当$E$有序连续范数时，这意味着每个无界不交保持映射$T:E\to F$有唯一分解$T=R+S$，其中$R$是有界不交保持算子，$S$是无界不交保持算子，在范数稠密理想上为零.对于$E=C(X)$，$X$是紧Hausdorff空间的情形，证明了每个保不交算子$T:C(X)\to F$在$C(X)$的范数稠密子格代数上是范数有界的，从而将$T$分解为一个有界的保不交算子和一个无界保不交算子的有限和，它们在序稠密理想上为零.
英文摘要：
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 $E\to F$ separate the points of $E$. Then we show that every disjointness preserving operator $T:E\to F$ is norm bounded on an order dense ideal. In case $E$ has order continuous norm, this implies that that every unbounded disjointness preserving map $T:E\to F$ 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 $T:C(X)\to F$ 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.
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