Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard splitW*-inclusions
Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard splitW*-inclusions
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阶乘 Stone-Weierstrass 猜想的解。
DOI：
10.1007/bf01388497
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发表时间：
1984
影响因子：
3.1
通讯作者：
R. Longo
中科院分区：
文献类型：
--
作者：
R. Longo
本文的主要结果是算子代数中一个长期存在的问题的正解，即在任意具有可分前元的因子M中，存在一个在M中奇异的近似有限维子因子R，即R'A M= c1。这个结果与RV Kadison’s Baton Rouge Problems[10]密切相关，在M是ii因子的情况下，S. Popa[16]得到了这个结果，而我们处理的是无穷因子。我们的结果的相关性通过其主要含义得到了更好的理解，这些含义多年来已经得到了认可（并且只需要无限因素的结果）。C*-代数理论中两个经典问题的正解（见[19，最后的注释]）将紧随其后。第一个应用表明，给定两个C*-代数A C B，且A可分离，A的任何因子状态q5可推广到B的一个因子状态。事实上，S. Sakai[20]的一个老论证，见[5,12]，证明了A是核的，或者更一般地说，7~(A)'含有奇异单射子因子的结果。第二个含义是弱Stone-Weierstrass猜想的正解，即如果A c B是可分离的c *-代数，并且A分离了B的因子状态，那么A= B。事实上Anderson和Bunce[-3]（参见[15]）已经证明了我们现在证明的结果。我们参考[2,4,8,9,11,18]关于非交换Stone-Weierstrass定理的早期发展。
The main result in this paper is the positive solution of a long-standing problem in Operator Algebras, namely the existence, in any factor M with separable predual, of an AFD (Approximately Finite Dimensional) subfactor R which is singular in M, ie R'A M= C 1. This result, which is closely related to some of the RV Kadison's Baton Rouge Problems [10], has been obtained by S. Popa [16] in case M is a II-factor, while we deal with infinite factors. The relevance of our result is better understood by its major implications, which have been recognized through the years (and need the result only for infinite factors). The positive solution of two classical problems in C*-algebra theory (see [19, final remarks]) will follow indeed. The first application shows that, given two C*-algebras A c B with A separable, any factor state q5 of A extends to a factor state of B. Indeed an old argument of S. Sakai [20], see [5, 12], proves the result if A is nuclear or, more generally, if 7~(A)'contains a singular injective subfactor. The second implication is the positive solution of the weak Stone-Weierstrass Conjecture, namely if A c B are separable C*-algebras and A separates the factor states of B, then A= B. Indeed Anderson and Bunce [-3](see also [15]) have shown that this follows by the result we are now proving. We refer to [2, 4, 8, 9, 11, 18] for the earlier developments on a non commutative Stone-Weierstrass theorem.