Derivations on Some Algebras of Measurable Operators Affiliated with Real W*-algebras of Type I
Аннотация
It is well known that every derivation on a von Neumann algebra is inner, which reflects the strong rigidity of these algebras. In contrast, for general C*-algebras there may exist non-inner derivations, indicating a more complicated and diverse algebraic structure. This fundamental difference has stimulated extensive research on derivations on various classes of operator algebras. In recent years, increasing attention has been paid to derivations defined on algebras of unbounded operators, in particular on algebras of measurable, locally measurable, and τ-measurable operators associated with von Neumann algebras. Such algebras arise naturally within the framework of noncommutative integration theory and provide a rich setting for extending classical results from the theory of bounded operators. In particular, a complete description of derivations on these algebras has been established in a number of works when they are associated with type I von Neumann algebras, demonstrating that under appropriate assumptions the derivations possess strong regularity properties and admit explicit representations. The present article is devoted to the development of a real analogue of the results described above. More precisely, derivations on algebras of measurable, locally measurable, and τ-measurable operators associated with real type I von Neumann algebras are investigated. By carefully adapting the methods from the complex case and taking into account the specific algebraic and topological features of real operator algebras, a complete characterization of all derivations on the algebras under consideration is obtained. These results generalize known theorems for complex von Neumann algebras to the real setting and contribute to a deeper understanding of derivations on algebras of unbounded operators associated with real operator algebras.