Short History

The theory of operator spaces grew out of the analysis of completely positive and completely bounded mappings. These maps were first studied on $C^*$-algebras, and later on suitable subspaces of $C^*$-algebras. For such maps taking values in ${B(\H)}$ representation and extension theorems were proved [Sti55], [Arv69], [Haa80], [Wit81], [Pau82]. Many of the properties shared by completely positive mappings can be taken over to the framework of operator systems [CE77]. Operator systems provide an abstract description of the order structure of selfadjoint unital subspaces of $C^*$-algebras. Paulsen's monograph [Pau86] presents many applications of the theory of completely bounded maps to operator theory. The extension and representation theorems for completely bounded maps show that subspaces of $C^*$-algebras carry an intrinsic metric structure which is preserved by complete isometries. This structure has been characterized by Ruan in terms of the axioms of an operator space [Rua88]. Just as the theory of $C^*$-algebras can be viewed as noncommutative topology and the theory of von Neumann algebras as noncommutative measure theory, one can think of the theory of operator spaces as noncommutative functional analysis.

This program has been presented to the mathematical community by E.G. Effros [Eff87] in his address to the ICM in 1986. The following survey articles give a fairly complete account of the development of the theory: [CS89], [MP94], [Pis97].