Riesz空间中的极大不相交系和表示理论

Maximal Disjoint Systems in Riesz Spaces and Representation Theory

设Ｅ是阿基米德Ｒｉｅｓｚ空间，有弱单位元ｅ和极大不相交系｛ｅｉ：ｉ∈Ｉ｝，其中每一个ｅｉ都是投影元素．由ｅｉ生成的主带记为Ｂ（ｅｉ）．本文考虑如下论述：（ａ）存在完全正则Ｈａｕｓｄｏｒｆ空间Ｘ，使Ｅ是Ｒｉｅｓｚ同构于Ｃ（Ｘ）；（ｂ）对每一个ｉ∈Ｉ，存在一个完全正则Ｈａｕｓｄｏｒｆ空间Ｘｉ使Ｂ（ｅｉ）是Ｒｉｅｓｚ同构于Ｃ（Ｘｉ）．我们证明（ａ）可推出（ｂ）．但其逆在一般情况下不成立．当（ｂ）成立时，我们得到一些与（ａ）等价的论述．

Let E be an Archimedean Riesz space possessing a weak unit e and a maximal disjoint system {e i :i∈ I} in which each e i is a projection element. The principal band generated by e i is denoted by B(e i). In this paper, consider the following statements: (a) there exists a completely regular Hausdorff space X such that E is Riesz isomorphic to C(X). (b) For every i∈ I there exists a completely regular Hausdorff space X i such that B(e i) is Riesz isomorphic to C(X i). We show that (a) implies (b). But the inverse is not true in general. Whenever (b) holds, we obtain some statements, each of which is equivalent to (a).