A Galois theory for finite factors
A Galois theory for finite factors
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有限因素的伽罗瓦理论
DOI：
10.3792/pja/1195524026
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发表时间：
1960
期刊：
影响因子：
--
通讯作者：
Z. Takeda
中科院分区：
文献类型：
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作者：
M. Nakamura;Z. Takeda
根据经典简单代数理论与连续有限因子的紧密类比，很自然地要求连续有限因子服从一种伽罗瓦理论。从字面上看，众所周知，I. M. Singer [4] 在这个方向上进行了尝试。本文将在下面的定理中对此进行尝试。如果 A 是标准作用于可分离希尔伯特空间 H 的连续有限因子，如果 G 是 A 的有限外自同构群，如果 B 是由 G 下所有不变元素组成的 A 的子因子，并且 B 的交换子 B’ 是有限的。然后，G 的所有子群和 B 到 A 之间的所有中间子因子的格在伽罗瓦对应关系下是对偶同构的，伽罗瓦对应关系携带子群 F 到在 F 下元素方式不变的中间子因子 C。预计 B’ 的假设可以从 G 的有限性得到证明，作者希望在下一次讨论中对此进行讨论。还需要注意的是，定理中 A 的连续性假设是多余的，因为离散有限因子没有非平凡的外自同构群。 1o 由于 A 标准作用于 H，因此对于任何 g 都有一个酉 u，例如 ( 1 ) =uxu*，其中 x 表示 g 对 x eA 的作用。在余下的部分中，为了方便起见，假设对应关系g-->u满足(2)u_=u*。值得注意的是，u 属于 B’，因为根据假设 x--x--uxu*。引理 1. 根据 (1)，g 给出 A’ 上的外自同构。如果 xeA’，则对于任何 aeA，(1) 和 (2) 意味着 ax auxu uaqxu uxa-lUg UgXU ( X a 表明 g 守恒 A’。因此 (1) 给出 A’ 上的自同构。如果它是内部的，则对于任何 xeA’ 存在一个酉 wA’，使得 xq--w*xw 或 uxu =w*xw，因此wuqx=xwuq 对于任何 xeA'，即 wu 与 A' 的每个元素都可交换。因此酉算子 w'=wu 属于 A。因此，由我们 A'，
According to a closed analogy between theories of classical simple algebras and continuous finite factors, it is natural to ask that a continuous finite factor obeys a kind of Galois theory. Literally, it is known that I. M. Singer [4 gave an attempt in this direction. This note will present a trial towards it in the following THEOREM. If A is a continuous finite factor acting standardly on a separable Hilbert space H, if G is a finite group of outer automorphisms of A, if B is the subfactor of A consisting of all elements invariant under G, and if moreover the commutor B’ of B is finite. Then, the lattices of all subgroups of G and of all intermediate subfactors between B to A are dually isomorphic under the Galois correspondence which carries a subgroup F to an intermediate subfactor C invariant under F in element-wise. It is expected that the assumption on B’ is provable from the finiteness of G for which the authors hope to discuss in the next occasion. It is also to be remarked that the continuity assumption on A in the theorem is superfluous since a discrete finite factor has no non-trivial group of outer automorphisms. 1o Since A acts standardly on H, there is a unitary u for any g such as ( 1 ) =uxu*, where x means the action of g on x eA. Throughout the remainder, for the sake of convenience, it is to be assumed that the correspondence g -->u satisfies (2) u_=u*. It is to be noticed that u belongs to B’, since x--x--uxu* by the assumption. LEMMA 1. By (1), g gives an outer automorphism on A’. If xeA’, then for any aeA, (1) and (2) imply ax auxu uaqxu uxa-lUg UgXU ( X a which shows that g conserves A’. Hence (1) gives an automorphism on A’. If it is inner, then there is a unitary wA’ such that xq--w*xw or uxu =w*xw for any xeA’, whence wuqx=xwuq for any xeA’, that is, wu commutes with every element of A’. Hence the unitary operator w’=wu belongs to A. Therefore, by we A’,