The fundamental group of the von Neumann algebra of a free group with infinitely many generators is ...
The fundamental group of the von Neumann algebra of a free group with infinitely many generators is ...
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具有无限多个生成元的自由群的冯诺依曼代数的基本群是......
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通讯作者：
Florin R Adulescu
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作者：
Florin R Adulescu
本文证明了具有无限多生成子的自由（非交换）群的von Neumann代数L（f1）的基本群F是R + nf0g。这推广了D. Voiculescu（[26], 27]）证明Q + nf0g包含在F中的结果，解决了自由群f1谐波分析中的一个经典问题。特别地，对于每一个s 24，存在索引为s的L（f1）子因子；1)。我们将使用Voiculescu论文中引入的非交换（量子）概率方法。冯·诺伊曼代数是由默里和冯·诺伊曼在三十年代初引入的，为量子物理学提供了一个框架。正如海森堡所表述的那样，量子化相当于用无限矩阵的非交换代数取代经典系统相空间上“可观察”函数的代数，或者更准确地说，是希尔伯特空间上的算子。根据定义，von Neumann代数是Hilbert空间上包含单位算子的有界算子的弱闭自伴随代数。然而，任何交换的冯·诺伊曼代数与测量空间上有界函数的代数都是*-同构的，而非交换代数的结构则要微妙得多。最简单的非交换冯诺依曼代数
In this paper we show that the fundamental group F of the von Neumann algebra L(F 1) of a free (noncommutative) group with innnitely many generators is R + nf0g. This extends the result of D. Voiculescu who previously proved ((26], 27]) that Q + nf0g is contained in F. This solves a classical problem in the harmonic analysis of the free group F 1. In particular it follows that there exists subfactors of L(F 1) with index s for every s 2 4; 1) .We will use the noncommutative (quantum) probabilistic approach introduced in Voiculescu's paper. Von Neumann algebras were introduced by Murray and von Neumann in the early thirties to provide a framework for quantum physics. As formulated by Heisenberg, quantization amounts to replacing the algebra of "observable" functions on the phase space of a classical system by a non-commuting algebra of innnite matrices , or more precisely, operators on a Hilbert space. By deenition , a von Neumann algebra is a weakly closed, self-adjoint algebra of bounded operators on a Hilbert space which contains the identity operator. Whereas any commutative von Neumann algebra is *-isomorphic to the algebra of bounded functions on a measure space , the structure of the non-commutative algebras is much more subtle. The simplest non-commutative von Neumann algebras