Abstract

We give an example of infinite dimensional central simple (von Neumann) regular algebra with outer derivations. Given the algebra \(S(M)\) of all measurable operators affiliated with the type II\({}_{1}\) hyperfinite factor \(M\) we construct its \(\ast\)-subalgebra \(\mathcal{R}_{\infty}\) which is dense in the measure topology. We prove that this algebra has a derivation which is discontinuous in the measure topology and hence is non-inner. We show that the algebra \(\mathcal{R}_{\infty}\) admits also a continuous (in the measure topology) derivation, implemented by a measurable operator, but which is still an outer derivation.

中文翻译：

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具有外导数的无穷维中央简单正则代数

摘要

我们给出了一个带有外部导数的无穷维中心简单（von Neumann）正则代数的示例。给定与类型II \（{} _ {1} \）超有限因子\ {M \ }关联的所有可测算子的代数\ {S（M）\ }，我们构造其\（\ ast \）-子代数\ { \ mathcal {R} _ {\ infty} \），在度量拓扑中比较密集。我们证明该代数的推导在度量拓扑中是不连续的，因此是非内在的。我们证明了代数\（\ mathcal {R} _ {\ infty} \）也接受由可测算子实现的连续（在度量拓扑中）推导，但仍是外推导。