Metric characterizations of isometries and of unital operator spaces and systems

Abstract

Abstract. We give some new characterizations of unitaries, isometries, unital operator spaces, unital function spaces, operator systems, C * -algebras, and related objects. These characterizations only employ the vector space and operator space structure (not mentioning products, involutions, or any kind of function on the space).

Weak∗ continuous states on Banach algebras

“…For this reason we state here another simple characterization. In the case of C * -algebras this characterization has been observed earlier by others and demonstrated to be useful [3].…”

Operator space and operator system analogs of Kirchberg's nuclear embedding theorem

“…[5,Theorem 4.4] provides an abstract characterization of operator systems among the matricially ordered vector spaces endowed with a linear involution (the adjoint map) and a distinguished element (the unit). An operator system is in particular an operator space, and an abstract characterization of operator systems among operator spaces with a unit has been given in [3].…”

Dual operator systems

“…We may replace ω by (1/2)(ω + ω * ), where ω * (v) := ω(v * ) for all v ∈ X. This is a weak* continuous functional, since the involution is weak* continuous on X by [7]. Since X + is a cone, we may take α = 0.…”

“…An abstract characterization of unital operator spaces may be found in [7]. The reader may also find metric characterizations of operator systems there, and the fact that the involution on a dual operator system is weak* continuous.…”