Integration theorems for gages and duality for unimodular groups
Integration theorems for gages and duality for unimodular groups
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量具的积分定理和幺模群的对偶性
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发表时间：
1959
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通讯作者：
W. Stinespring
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作者：
W. Stinespring
附属于 a.关于 (t 上的“量具”。在目前的工作中，我们研究了普通积分理论的某些部分的类似物，而 Segal 没有机会在 [10] 中发展这些部分，然后我们将量具空间理论应用于调和分析和幺模群的对偶性。在普通测度论中构造积分的最重要方法之一是从某些有界函数代数上的正线性泛函开始。在 ?1 中，相应的构造 考虑了量具的数量。定理 1 给出了有界算子的自伴代数上的正中心线性泛函在其生成的 W* 代数上产生一个量具的条件。该定理在 7 中应用于乘积规范的构造，在 9 中应用于单模群的双规范构造。自然出现了关于非中心正线性泛函的推广可以说些什么的问题。一个 特殊情况在第2 节中处理。在[10]中，几乎处处收敛（n.e.）是为量规空间定义的。在?3中，引入了度量收敛的概念。人们发现它与收敛有通常的关系。和平均收敛性。在?4 中，几个主导收敛定理与 Fatou 引理的一个版本一起被证明。测度收敛最常用于概率 空间。在[10]之后的工作中，Segal 使用了概率量空间上自伴算子的概率收敛概念，其定义与我们的测度收敛不同。然而，两人是
affiliated with a. with respect to a "gage" on (t. In the present work, we investigate analogues of certain portions of ordinary integration theory which Segal did not have occasion to develop in [10], and then we apply the theory of gage spaces to harmonic analysis and duality of unimodular groups. One of the most important ways of constructing an integral in ordinary measure theory is by starting from a positive linear functional on some algebra of bounded functions. In ?1, the corresponding construction of gages is considered. Theorem 1 gives conditions under which a positive central linear functional on a self-adjoint algebra of bounded operators yields a gage on the W*-algebra it generates. This theorem is applied in ?7 to the construction of product gages and in ?9 to the construction of the dual gage for a unimodular group. The question of what can be said about the extension of noncentral positive linear functionals naturally arises. A special case is dealt with in ?2. In [10], convergence nearly everywhere (n.e.) is defined for gage spaces. In ?3, a notion of convergence in measure is introduced. It is found to have the usual relations to convergence n.e. and mean convergence. In ?4, several dominated convergence theorems are proved together with a version of Fatou's lemma. Convergence in measure is most often used in probability spaces. In later work than [10], Segal has used a notion of convergence in probability for self-adjoint operators on a probability gage space, which is defined differently from our convergence in measure. However, the two are