CONVERGENCE IN MEASURE AND RELATED RESULTS IN FINITE RINGS OF OPERATORS
CONVERGENCE IN MEASURE AND RELATED RESULTS IN FINITE RINGS OF OPERATORS
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算子有限环测度收敛性及相关结果
DOI：
10.1090/s0002-9947-1967-0212581-7
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发表时间：
1967
影响因子：
1.3
通讯作者：
A. Padmanabhan
中科院分区：
文献类型：
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作者：
A. Padmanabhan
介绍。 Segal 在 [6] 中结合量子力学、算子代数和群调和分析的研究奠定了非交换积分理论的基础。在上述学科中，出现了与常规测度空间上的可测函数（非交换）类似的系统，此类系统的最简单实例是所谓的 I1 型因子，例如，所有复数 n x n 矩阵与迹一起组成的环。如果迹被归一化以便在恒等算子上假设值统一，则该系统成为在由 n 个原子生成的概率空间上复杂随机变量的非交换模拟。令 G 为局部紧幺模群（例如，对于某些固定 n，所有实数 n x n 非奇异矩阵组成的群），tu 是 G 上的 Haar 测度，A 是 G 上所有有界可积函数的代数，乘法定义为卷积。 (A, ,uk) 对提供了开头描述的排序示例。设H=L2(G,...,u)，其中G是所有变换cx+d、c、d有理数的群，JL是G上的计数测度。对于H中的f、g，G中的a、b，令U0定义如下：Ujf=g，其中g(b)=f(ba)。令 M 为所有有界算子的环，对于所有 a，其与 Ua 交换。 H 中的每个有界运算符 A 都可以用有界数值矩阵 A | 的形式表示。 , G 中的 a, b。对于 M 中的任意 A，设 (A)= e，（e 是 G 的恒等式）。让我表示恒等运算符。对于 M 中的任意 C、D，r(CD) = r(DC) 和 r(I) = 1。M 是 I1l 类型的因子，是非原子概率空间上有界复数随机变量的非交换模拟。 I1 型因子的一个特殊情况，称为“近似有限因子”，在费米-狄拉克量子化理论中以自然的方式出现，如论文 [7]、[8] 和 [9] 中所述。西格尔在最一般的背景下提出的整合理论可以概括为这样。给出了运算符环，以及在其某些元素上定义的跟踪。然后通过适当的收敛概念将轨迹扩展到更广泛的类别。并且，对于这个扩大的系综（其元素称为可积），获得了标准测度理论结果的类似物。在任意量规空间中，Segal 引入了可测算子、几乎处处收敛的概念，并对其进行了相当广泛的研究。
Introduction. The foundations of a noncommutative integration theory were laid by Segal in [6], in connection with investigations in quantum mechanics, operator algebras and harmonic analysis on groups. In the aforesaid disciplines, there arise systems which are (noncommutative) analogues of measurable functions on a conventional measure space, the simplest instances of such systems being the so-called factors of type I1, e.g., the ring of all complex n x n matrices together with the trace. If the trace is normalized so as to assume the value unity, on the identity operator, then this system becomes the noncommutative analogue of complex random variables, on a probability space generated by n atoms. Let G be a locally compact unimodular group (such as, for some fixed n, the group of all real n x n nonsingular matrices), tu the Haar measure on G, and A the algebra of all bounded, integrable functions on G, with multiplication defined as convolution. The pair (A, ,uk) provides an example of the sort described in the beginning. Let H=L2(G, ,u) where G is the group of all transformations cx+ d, c, d rational, and JL is the counting measure on G. Forf, g in H, and a, b in G, let U0 be defined thus: Ujf=g, where g(b) =f(ba). Let M be the ring of all bounded operators which commute with Ua for all a. Every bounded operator A in H is representable in the form of a bounded numerical matrix A | , a, b in G. For any A in M, set (A)= e, (e being the identity of G). Let I denote the identity operator. For arbitrary C, D in M, r(CD) = r(DC), and r(I) = 1. M, a factor of type I1l, is the noncommutative analogue of bounded, complex random variables on a nonatomic probability space. A special case of a type I1 factor, bearing the appellation "approximately finite factor," arises in a natural way, in the theory of Fermi-Dirac quantization as described in the papers [7], [8], and [9]. The integration theory, developed by Segal in the most general setting, may be epitomized thus. A ring of operators, with a trace defined on some elements thereof is given. The trace is then extended to a wider class via suitable convergence concepts. And, for this enlarged ensemble (whose elements are called integrable) analogues of standard measure-theoretic results are obtained. In an arbitrary gage space, Segal introduced, and made a fairly extensive study of, the concepts of measurable operators, convergence nearly everywhere, and