The spectral theory of distributive continuous lattices
The spectral theory of distributive continuous lattices
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DOI：
10.1090/s0002-9947-1978-0515540-7
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发表时间：
1978-12
影响因子：
1.3
通讯作者：
K. Hofmann;J. Lawson
中科院分区：
文献类型：
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作者：
K. Hofmann;J. Lawson
本文研究了分配连续格的谱（即具有壳-核拓扑的素元集）的各种性质。证明了谱总是局部拟紧sober空间，反之，局部拟紧sober空间的开集格是连续格。代数格是连续格的一个特殊子类，本文讨论了代数格的谱的一些特殊性质.将补片拓扑的概念从代数格推广到连续格，给出了补片拓扑紧的充要条件。格的谱理论的目的是将格L表示为拓扑空间X的开集的格。谱理论的环和代数实际上减少到这种情况下，鉴于事实上，在大多数情况下，一个考虑格的环（或代数）理想，然后发展的谱理论的格。(The由于理想产物不是交叉点的事实而引起的偶然的复杂性已经在别处处理，例如（4）。所有环（或代数）理想的格形成一种特殊的连续格，即代数格。它应该是这样的情况下，然而，更一般的连续格出现在研究某些对象赋予了代数和拓扑结构。事实上，第一作者已经表明，在一个研讨会报告中使用的概念佩德森的理想，封闭的理想的C*-代数总是形成一个分配连续格方面的交叉。在这一点上，连续格在这样的背景下出现的范围有多广，在很大程度上是一个未知的海洋。我们证明了分配连续格的谱是局部拟紧sobriety空间（sobriety的定义见2.6）。这意味着，例如，证明了C*-代数的闭双边素理想空间在壳-核拓扑中是局部拟紧的. (This通常用不同的方法对本原理想证明）。另一方面，拓扑学后果的问题
In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass of continuous lattices and the special proper- ties of their spectra are treated. The concept of the patch topology is extended from algebraic lattices to continuous lattices, and necessary and sufficient conditions for its compactness are given. The spectral theory of lattices serves the purpose of representing a lattice L as a lattice of open sets of a topological space X. The spectral theory of rings and algebras practically reduces to this situation in view of the fact that for the most part one considers the lattice of ring (or algebra) ideals and then develops the spectral theory of that lattice. (The occasional complications due to the fact that ideal products are not intersections have been dealt with elsewhere, e.g. (4).) The lattice of all ring (or algebra) ideals forms a particular kind of continuous lattice, namely an algebraic lattice. It should be the case, however, that more general continuous lattices arise in the study of certain objects endowed with both an algebraic and a topological structure. Indeed the first author has shown in a seminar report using the concept of Pedersen's ideal that the closed ideals of a C*-algebra always form a distributive continuous lattice with respect to intersection. How widely continuous lattices occur in such contexts is, at this point, a largely uncharted sea. We show that the spectrum of a distributive continuous lattice is a locally quasicompact sober space (see 2.6 for the definition of sobriety). This implies, e.g., that the space of closed two sided prime ideals of a C*-algebra is locally quasicompact in the hull-kernel topology. (This is usually proved for primitive ideals by different methods.) On the other hand, the question of what topological consequences follow