Various averaging operations onto subalgebras
Various averaging operations onto subalgebras
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对子代数的各种平均运算
DOI：
10.1215/ijm/1255455460
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发表时间：
1959
影响因子：
0.6
通讯作者：
Chandler Davis
中科院分区：
文献类型：
--
作者：
Chandler Davis
本文所讨论的运算是定义在复数上的自伴代数上的函数，其值在它的子代数中（子代数是任意的）。它们被称为“平均”，因为定义它们的形式性质（定义2.1）被I的条件期望所拥有。E.西格尔如果子代数只是常数，则平均运算就是状态;如果子代数是整个代数，则唯一的平均运算就是恒等式。在整个代数是交换的情况下，G. Birkhoff [2]和J. L.凯利[10]。另一类操作（我要求没有新奇，除了名称）介绍，并提出基本属性，在3。我开始研究这个类的原因有些不同，但结果证明它与平均运算密切相关--事实上，它是一个子类，而且这个子类的行为最不像阿贝尔情形中出现的经典条件期望。本文的主要结果是用这两种特殊类型表示的任意平均运算的表达式。平均操作对厄米算子谱的影响是6-7的主题。定理7.2是哈代、利特尔伍德和PSlya的定理的简单推广，可能有独立的意义。本文中的所有代数都是有限维的。推广的一些结果，以任意杨诺依曼代数的想法，这使我经常表达的东西，代数，交换子等，而有些证明只使用矩阵会更短一些。[1959年3月31日]刻画非交换条件期望的程序，类似于伯克霍夫、莫伊和其他人在交换情况下的工作，是由M。中村和他的同事;特别见[13]。我很遗憾，我在写这篇论文时对这部作品一无所知。他们的结果处理无限维的情况;他们似乎不包含我这里的主要结果作为专门化。我感谢公关。Halmos就本文的主题进行了几次有益的对话。
The operations dealt with in this paper are functions defined on a selfadjoint algebra over the complex numbers and with values in a subalgebra of it. (The subalgebra is arbitrary.) They are called "averaging" because the formal properties which define them (Definition 2.1) are possessed by the conditional expectations of I. E. Segal. If the subalgebra is just the constants, the averaging operations are exactly the states; if the subalgebra is the whole algebra, the only averaging operation is the identity. In case the whole algebra is commutative, the study of averagings has been carried very far by G. Birkhoff [2] and J. L. Kelley [10]. Another class of operations (for which I claim no novelty except the name) is introduced, and elementary properties set forth, in 3. I had begun studying this class for somewhat different reasons, but it turns out to be closely related to the averaging operations--indeed, to be a subclass, and that subclass which behaves least like the classical conditional expectations which occur in the abelian case. The main result of this paper is the expression (in 4) of an arbitrary averaging operation in terms of these two special types. The effect of averaging operations on the spectrum of a hermitian operator is the subject of 6-7. Theorem 7.2, a simple extension of a theorem of Hardy, Littlewood, and PSlya, may have independent interest. All the algebras in this paper are finite-dimensional. Extension of some of the results to arbitrary yon Neumann algebras is the idea which leads me often to express things in terms of algebras, commutors, etc., when some proofs would be a little shorter using only matrices. [Added March 31, 1959. The program of characterizing noncommutative conditional expectations, in analogy to the work of Birkhoff, Moy, and others in the commutative case, was initiated by M. Nakamura and his colleagues; see especially [13]. I regret that I was in ignorance of this work when I wrote the present paper. Their results deal with the infinite-dimensional case; they do not seem to contain my main results here as specializations.] I thank P. R. Halmos for several helpful conversations on the subject of this paper.