一类可测函数环上的结构空间及相关问题

Soumajit Dey, Sudip Kumar Acharyya, Dhananjoy Mandal
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引用次数: 0

摘要

如果在可测空间$(X,\mathcal{A})$上定义的实值$\mathcal{A}$可测函数的环$S(X,\mathcal{A})$的特征函数$\chi_{E}\inS(X,\mathcal{A})$称为$\chi$环。在$X$上所有具有斯通拓扑$\tau$的$\mathcal{A}$超滤波器的集合$\mathcal{U}_X$与在$S(X,\mathcal{A})$中所有具有赫尔核拓扑$\tau_S$的最大理想的集合$\mathcal{M}_X$的一个适当的同调空间是同构的。我们认识到,当且仅当 $S(X,\mathcal{A})$ 是一个格尔芬环时,$(\mathcal{U}_X,\tau)$ 与$(\mathcal{M}_S,\tau_S)$ 是同构的。我们进一步观察到,当且仅当这个环中的每个理想都是 $\mathcal{Z}_S$ 理想时,$S(X,\mathcal{A})$ 是冯-诺伊曼正则环;当且仅当这个环中的每个最大理想都是 $\mathcal{Z}_S$ 理想时,$S(X,\mathcal{A})$ 是格尔方环。在集合$S(X,\mathcal{A})$上引入了一对拓扑$u_\mu$-拓扑和$m_\mu$-拓扑,并研究了它们的一些性质。
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Structure spaces and allied problems on a class of rings of measurable functions
A ring $S(X,\mathcal{A})$ of real valued $\mathcal{A}$-measurable functions defined over a measurable space $(X,\mathcal{A})$ is called a $\chi$-ring if for each $E\in \mathcal{A} $, the characteristic function $\chi_{E}\in S(X,\mathcal{A})$. The set $\mathcal{U}_X$ of all $\mathcal{A}$-ultrafilters on $X$ with the Stone topology $\tau$ is seen to be homeomorphic to an appropriate quotient space of the set $\mathcal{M}_X$ of all maximal ideals in $S(X,\mathcal{A})$ equipped with the hull-kernel topology $\tau_S$. It is realized that $(\mathcal{U}_X,\tau)$ is homeomorphic to $(\mathcal{M}_S,\tau_S)$ if and only if $S(X,\mathcal{A})$ is a Gelfand ring. It is further observed that $S(X,\mathcal{A})$ is a Von-Neumann regular ring if and only if each ideal in this ring is a $\mathcal{Z}_S$-ideal and $S(X,\mathcal{A})$ is Gelfand when and only when every maximal ideal in it is a $\mathcal{Z}_S$-ideal. A pair of topologies $u_\mu$-topology and $m_\mu$-topology, are introduced on the set $S(X,\mathcal{A})$ and a few properties are studied.
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