Quasi-multipliers and algebrizations of an operator space

Abstract

Let $X$ be an operator space, let $ϕ$ be a product on $X$, and let $(X,ϕ)$ denote the algebra that one obtains. We give necessary and sufficient conditions on the bilinear mapping $ϕ$ for the algebra $(X,ϕ)$ to have a completely isometric representation as an algebra of operators on some Hilbert space. In particular, we give an elegant geometrical characterization of such products by using the Haagerup tensor product. Our result makes no assumptions about identities or approximate identities. Our proof is independent of the earlier result of Blecher-Ruan-Sinclair that solved the case when the algebra has an identity of norm one, and our result is used to give a simple direct proof of this earlier result. We also develop further the connections between quasi-multipliers of operator spaces, and shows that the quasi-multipliers of operator spaces coincide with their $C^*$-algebraic counterparts.