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RESEARCH PRODUCT

Locally Convex Quasi C*-Algebras and Their Structure

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Throughout this chapter \({{\mathfrak A}}_{\scriptscriptstyle 0}[\| \cdot \|{ }_{\scriptscriptstyle 0}]\) denotes a unital C*-algebra and τ a locally convex topology on \({{\mathfrak A}}_{\scriptscriptstyle 0}\). Let \(\widetilde {{{\mathfrak A}}_{\scriptscriptstyle 0}}[\tau ]\) denote the completion of \({{\mathfrak A}}_{\scriptscriptstyle 0}\) with respect to the topology τ. Under certain conditions on τ, a subspace \({\mathfrak A}\) of \(\widetilde {{{\mathfrak A}}_{\scriptscriptstyle 0}}[\tau ]\), containing \({{\mathfrak A}}_{\scriptscriptstyle 0}\), will form (together with \({{\mathfrak A}}_{\scriptscriptstyle 0}\)) a locally convex quasi *-algebra \(({\mathfrak A}[\tau ],{{\mathfrak A}}_{\scriptscriptstyle 0})\), which is named locally convex quasi C*-algebra. Examples and basic properties of such algebras are presented. So, let \({{\mathfrak A}}_{\scriptscriptstyle 0}[\| \cdot \|{ }_{\scriptscriptstyle 0}]\) and τ be as before, with {pλ}λ ∈ Λ a defining family of seminorms for τ. Suppose that τ satisfies the properties: (T1) \({{\mathfrak A}}_{\scriptscriptstyle 0}[\tau ]\) is a locally convex *-algebra with separately continuous multiplication. (T2) τ ≼∥⋅∥0.