In this paper, we prove a Krein space J-representation theorem for local \(\alpha \)-completely positive maps on locally \(C^*\)-algebras. Using this representation, we construct a Krein space J-representation associated with a pair of two maps \((\varphi ,\Phi )\) where \(\varphi \) is a local \(\alpha \)-completely positive map on a locally \(C^*\)-algebra and \(\Phi \) is a \(\varphi \)-map. Also, we discuss the minimality of Krein space J-representations, and as an application, we establish the Radon–Nikodým theorem for local \(\alpha \)-completely positive maps.

在本文中，我们证明了局部\(\alpha \) -在局部\(C^*\) -代数上的完全正映射的Kerin空间J -表示定理。使用这种表示，我们构建了一个与一对两个映射\((\varphi ,\Phi )\)相关的 Kerin 空间J -表示，其中\(\varphi \)是一个局部的\(\alpha \) -完全正映射在本地\(C^*\) -代数和\(\Phi \)是\(\varphi \) -map。此外，我们讨论了 Kerin 空间J表示的极小性，并且作为一个应用，我们建立了局部\(\alpha \)的 Radon–Nikodým 定理-完全正面的地图。