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RESEARCH PRODUCT

Commutative Partial O*-Algebras

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This chapter is devoted to the integrability of commutative partial O*-algebras. Three notions of weak commutativity, commutativity and strong commutativity of an O*-vector space are defined and investigated. In Section 3.1, we analyze the relation between the integrability of weakly commutative O*-vector space M and the commutativity of the von Neumann algebra (M w ′ )′. In Section 3.2, we study the integrable extensions of partial O*-algebras. In Section 3.3, we describe another explicit example, namely, the partial O*-algebra M[S, T] generated by two weakly commuting symmetric operators S and T defined on a common dense domain in a Hilbert space. In particular, we investigate in detail the structure, the integrability and the integrable extensions of this commutative partial O*-algebra. Section 3.4 is devoted to the construction of nonintegrable commutative self-adjoint O*-algebras: Given a properly infinite von Neumann algebra A on a separable Hilbert space H, there exists a commutative self-adjoint O*-algebra P(S, T) on a dense subspace D in H, such that S n and T n are essentially self-adjoint for each n ∈ ℕ and (P(S,T) w ′ )′ = A.