Locally convex quasi C*-algebras and noncommutative integration

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 L^2-spaces.