Abstract: 
\def\La{{\mit\Lambda}}%
We deal with dual complementors on complemented
topological (non-normed) algebras and give some characterizations
of a dual pair of complementors for some classes of complemented
topological algebras. The study of dual complementors shows their
deep connection with dual algebras. In particular, we refer to
Hausdorff annihilator locally $C^*$-algebras and to proper
Hausdorff orthocomplemented locally convex $H^*$-algebras. These
algebras admit, by their nature, the same type of dual pair of
complementors. Dual pairs of complementors are also obtained on
their closed 2-sided ideals or even on particular 1-sided ideals.
If $(\perp_l ,\perp_r )$ denotes a pair of complementors on a
complemented algebra, then through the notion of a $\perp_l$
(resp. $\perp_r)$-projection, we get a structure theorem (analysis
via minimal 1-sided ideals) for a semisimple annihilator left
complemented $Q'$-algebra. Actually, such an algebra contains a
maximal family, say $(x_i )_{i\in \La}$, of mutually orthogonal
minimal $\perp_l$-projections and the respective minimal ideals
(factors of the analysis) are the $Ex_i$ and $x_i E$, $i\in \La$.
As a consequence, an analysis is given for a certain locally
$C^*$-algebra. In this case, the respective $x_i$'s are, in
particular, projections in both (left and right) complementors.

1. Department of Mathematics
   University of Athens
   Panepistimiopolis, Athens 15784
   Greece