The main aim of this paper is to generalize the classical concept of a positive operator, and to develop a general extension theory, which overcomes not only the lack of a Hilbert space structure, but also the lack of a normable topology. The concept of anti-duality carries an adequate structure to define positivity in a natural way, and is still general enough to cover numerous important areas where the Hilbert space theory cannot be applied. Our running example – illustrating the applicability of the general setting to spaces bearing poor geometrical features – comes from noncommutative integration theory. Namely, representable extension of linear functionals of involutive algebras will be governed by their induced operators. The main theorem, to which the vast majority of the results is built, gives a complete and constructive characterization of those operators that admit a continuous positive extension to the whole space. Various properties such as commutation, or minimality and maximality of special extensions will be studied in detail.

中文翻译：

---

对偶对算子：广义 Krein-von Neumann 扩展

本文的主要目的是推广正算子的经典概念，并发展一般可拓理论，它不仅克服了希尔伯特空间结构的缺乏，而且克服了可规范拓扑的缺乏。反二元性的概念具有足够的结构来以自然的方式定义积极性，并且仍然足够概括以涵盖希尔伯特空间理论无法应用的许多重要领域。我们的运行示例——说明一般设置对几何特征较差的空间的适用性——来自非对易积分理论。即，对合代数的线性泛函的可表示扩展将由它们的诱导算子控制。建立绝大多数结果的主要定理，给出了那些允许对整个空间进行连续正扩展的算子的完整和建设性的表征。将详细研究各种性质，例如交换，或特殊扩展的最小和最大。