Multiplicative Structures

For an abstract $C^*$-algebra the GNS construction provides a concrete representation of its elements as bounded operators on a Hilbert space. For non-selfadjoint algebras there is, hitherto, no analogue in the framework of classical functional analysis. But endowed with an operator space structure (which is compatible with the multiplicative structure), these non-selfadjoint algebras do have a representation in some $B({\mathcal{H}})$ ( theorem of Ruan type for operator algebras).

The so-called operator modules (over algebras) are also characterized by Axioms of Ruan type; here, matrices whose entries are algebra elements take the place of the scalar ones. The corresponding morphisms are the completely bounded module homomorphisms , the most important properties of which make their appearance in Representation, Decomposition and Extension Theorems.