Locally convex quasi C*-algebras and noncommutative integration

In this paper 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 and we focus our attention on the so-called {\em locally convex quasi C*-algebras}. We show, in particular, that any strongly *-semisimple locally convex quasi C*-algebra $(\X,\Ao)$, can be represented in a class of noncommutative local $L^2$-spaces.