Local operator spaces, unbounded operators and multinormed C∗-algebras

In this paper we propose a representation theorem for local operator spaces which extends Ruan’s representation theorem for operator spaces. Based upon this result, we introduce local operator systems which are locally convex versions of the operator systems and prove Stinespring theorem for local operator systems. A local operator C∗-algebra is an example of a local operator system. Finally, we investigate the injectivity in both local operator space and local operator system senses, and prove locally convex version of the known result by Choi and Effros, that an injective local operator system possesses unique multinormed C∗-algebra structure with respect to the original involution and matrix topology. © 2008 Elsevier Inc. All rights reserved.