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Topologies on Partial O*-Algebras

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In this chapter, we introduce some basic locally convex topologies on partial O*-algebras and we establish general properties of these topologies. In Section 4.1, we compare the graph topologies induced by different O-families on the same domain (and the corresponding families of bounded subsets). In the case where the domain D M of an O-family M is a (quasi-) Frechet space, the structure of bounded subsets in D M can be described in a rather explicit way. Section 4.2 and Section 4.3 are devoted to the topologization of (partial) O*-algebras. Section 4.2 deals with locally convex topologies, the so-called uniform topologies τ u , τ u , τ * u and quasiuniform topologies τ qu , and Section 4.3 deals with the inductive limit topologies called the ρ-topology τ ρ and the λ-topology τ λ , and these topologies are compared. All of these topologies are generalizations of the operator norm topology in C*- algebra theory. Section 4.4 is devoted to the investigation of the relations among these topologies (τ u , τ u , τ * u , τ ρ , τ λ) and the weak and σ-weak topologies τ w , τ σw and the strong, strong* and σ-strong topologies τ s , τ s*, τ σs introduced in Section 2.5.