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RESEARCH PRODUCT

Partial O*-Algebras

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This chapter is devoted to the investigation of partial O*-algebras of closable linear operators defined on a common dense domain in a Hilbert space. Section 2.1 introduces of O- and O*-families, O- and O*-vector spaces, partial O*-algebras and O*-algebras. Partial O*-algebras and strong partial O*-algebras are defined by the weak and the strong multiplication. Section 2.2 describes four canonical extensions (closure, full-closure, adjoint, biadjoint) of O*-families and defines the notions of closedness and full-closedness (self-adjointness, integrability) of O*-families in analogy with that of closed (self-adjoint) operators. Section 2.3 deals with two weak bounded commutants M′w and M′qw of an O*-family M, that play an important role for the study of (partial) O*-algebras. The relation between these commutants and the self-adjointness of M is investigated. In Section 2.4, we investigate induced extensions of partial O*-algebras, which play a crucial role in unbounded generalizations of von Neumann algebras developed in Section 2.5 and in the study of integrable extensions of partial O*-algebras in Section 3.2. In Section 2.5, the notion of (partial) GW*-algebras is defined and studied by considering the strong*-closure and the unbounded bicommutant of an O*-family. This is a natural unbounded generalization of the notion of von Neumann algebras. Section 2.6 is devoted to the construction of the partial O*-algebra M[T] generated by a single symmetric operator T. In particular, we investigate the bounded commutants and the integrability of M[T]. This last section is quite technical and may be skipped at first reading.