Simple injective subfactors
Simple injective subfactors
复制标题
简单内射子因子
DOI：
10.1016/0001-8708(87)90051-x
复制
发表时间：
1987
影响因子：
1.7
通讯作者：
R. Longo
中科院分区：
文献类型：
--
作者：
R. Longo
本文证明了每个具有可分前对偶的无穷因子M都包含一个决定M的同构的内射子因子R：如果4，$：M+ N是到上的同构且4 1 R= Ic/1 R，则I $= $.注意，根据Connes定理[4]，可以用a-弱稠密矩阵C*-子代数代替R。然而，不是R的每个同构都延伸到M（如果我们预先指定范围），尽管它总是延伸（根据内射性的定义）到M的完全正映射。我们的结果是一个更强定理的推论，即R在M中是单的，因为M在L '（M）上的自然左和右作用对R的限制是共同不可约的，我们将在下面解释。这个结构是全新的。即使在例子中，也不知道存在这样的子因子，也不清楚如何以我们的方式来构造它。此外，我们的构造是规范的，在M的某个状态的初始选择上（和I型子因子，但这是M中唯一确定的模内共轭），并通过分析vonNeumann代数包含的联合模结构得到（这最终依赖于自然锥理论[29，32，1，2，121]）。为了清楚起见，我们分别更详细地概述了这项工作的两个主要方面。
In this paper we shall show that every infinite factor M with separable predual contains an injective subfactor R that determines the isomorphisms of M: if 4, $: M+ N are isomorphisms onto and 4 1 R= Ic/1 R then I $= $. Note that, by Connes theorem [4], one can replace R by a a-weakly dense matricial C*-subalgebra. However, not every isomorphism of R extends to M (if we preassign the range) although it always extends (by the definition of injectivity) to a completely positive map of M. Our result is a corollary of a stronger theorem to the effect that R is simple in M in the sense that the restrictions to R of the natural left and right actions of M on L’(M) are jointly irreducibly, as we explain below. This structure is completely new. Even in the examples no such subfactor was known to exist nor is it clear how to construct it other than in our way.Moreover our construction is canonical, upon the initial choice of a certain state of M (and a type I subfactor, but this is uniquely determined modulo inner conjugacy in M), and is obtained by an analysis of the joint modular structure for an inclusion of von Neumann algebras (which ultimately relies on the theory of natural cones [29, 32, 1, 2, 121). For the sake of clarity we outline separately in more detail the two main strands of this work.