The modular operator is a crucial concept in the theory of von Neumann algebras that arises from the Tomita-Takesaki theory, acting on a given von Neumann algebra associated with a cyclic vector. It provides a systematic way to understand the structure and relationships of the algebra's elements, particularly in terms of modular conjugation and the modular flow. This operator plays a significant role in various applications, including statistical mechanics, quantum field theory, and the study of KMS states.
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