Representations of a module 𝔛 over a *-algebra $𝔄_{#}$ are considered and some related seminorms are constructed and studied, with the aim of finding bounded *-representations of $𝔄_{#}$.
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We continue the analysis undertaken in a series of previous papers on structures arising as completions of C*-algebras under topologies coarser that their norm topology and we focus our attention on the so-called locally convex quasi C*-algebras. We show, in particular, that any strongly *-semisimple locally convex quasi C*-algebra (𝔛,𝔄₀) can be represented in a class of noncommutative local L²-spaces.
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Non-commutative $L^{p}$-spaces are shown to constitute examples of a class of Banach quasi *-algebras called CQ*-algebras. For p ≥ 2 they are also proved to possess a sufficient family of bounded positive sesquilinear forms with certain invariance properties. CQ*-algebras of measurable operators over a finite von Neumann algebra are also constructed and it is proven that any abstract CQ*-algebra (𝔛,𝔄₀) with a sufficient family of bounded positive tracial sesquilinear forms can be represented as a CQ*-algebra of this type.
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