Dual complementors in topological algebras
Volume 67 / 2005
Abstract
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 $(x_i )_{i\in {\mit\Lambda}}$, of mutually orthogonal minimal $\perp_l$-projections and the respective minimal ideals (factors of the analysis) are the $Ex_i$ and $x_i E$, $i\in {\mit\Lambda}$. 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.
