ISOMORPHISMS OF FINITE TYPE II RINGS OF OPERATORS
ISOMORPHISMS OF FINITE TYPE II RINGS OF OPERATORS
复制标题
有限II型算子环的同构
DOI：
10.2307/1970018
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发表时间：
1956
影响因子：
4.9
通讯作者：
J. Feldman
中科院分区：
文献类型：
--
作者：
J. Feldman
保留*运算的两个算子环之间的环同构，通过限制推导出两个环中幺正算子群的同构。前段时间，H. a . Dye提出了算子环的环，*结构是否完全由其酉群的群结构决定的问题。他在[2]中证明了，对于某一类算子环，环，*结构是由这个群结构加上关于群的某些拓扑信息决定的。现在，有一种简单的方法把酉算子和投影联系起来；也就是说，将P的投影映射到酉I - 2P上。这种对应关系是环上所有算子的投影和环上所有二阶酉之间的1-1映射。Dye和I. Kaplansky都能够证明，在II型因子的情况下，这种对应关系可以将原来关于酉群的问题简化为环，*结构是否由环上投影集合中的序和正交关系决定的问题。在此之前，作者已经提出了这个问题，并将冯·诺伊曼的技术和结果应用于正则环和连续几何，得到了以下定理：如果两个不含2齐次和算子的半有限环的投影格通过一个保持正交性的同构是同构的，那么这个格同构是由一个环*同构引起的。戴伊后来独立地证明了这个定理，并且实际上能够免除半有限的假设。接下来的问题是从这里回到一般情况下关于酉群的定理。戴伊成功地对因子（至少是那些定理成立的因子：那些不属于12n型的因子）做了这样的处理。本文首先证明了有限型II算子环的点阵投影定理（在给定von Neumann连续几何定理的情况下，证明相对简单），然后证明了这类算子环的酉群定理。2. 预赛
A ring isomorphism between two rings of operators which preserves the * operation induces, by restriction, an isomorphism of the groups of unitary operators in the two rings. Some time ago, H. A. Dye raised the question whether the ring, * structure of a ring of operators was completely determined by the group structure of its unitary group. He proved in [2] that, for a certain class of rings of operators, the ring, * structure was determined by this group structure plus certain topological information about the group. Now, there is a simple way of associating unitary operators with projections; namely, send the projection P to the unitary I - 2P. This correspondence is a 1-1 map between all projections in a ring of operators and all unitaries of order 2 in the ring. Both Dye and I. Kaplansky were able to show that in the case of a factor of type II, this correspondence made it possible to reduce the original question about unitary groups to the question whether the ring, * structure was determined by the order and orthogonality relations in the set of projections in the ring. Previously to this, the present author had raised this very question, and, by applying J. von Neumann's techniques and results on regular rings and continuous geometries, had gotten the following theorem: if the lattices of projections of two semifinite rings of operators without 2-homogeneous summands are isomorphic via an isomorphism which preserves orthogonality, then the lattice isomorphism is induced by a ring, * isomorphism. Dye later proved the theorem independently, and was in fact able to dispense with the assumption of semifiniteness. Next came the problem of getting from here back to the theorem about unitary groups, in more general cases. Dye succeeded in so doing for factors (at least, those factors in which the theorem is true: those not of type 12n). In the present paper, we first prove the lattice of projections theorem for finite type II rings of operators (where the proof is relatively simple, given von Neumann's continuous geometry theorems), and then prove the unitary group theorem for this same class of rings of operators. 2. Preliminaries