Dual operator systems

Abstract

We characterize weak* closed unital vector spaces of operators on a Hilbert space H. More precisely, we first show that an operator system, which is the dual of an operator space, can be represented completely isometrically and weak* homeomorphically as a weak* closed operator subsystem of B(H). An analogous result is proved for unital operator spaces. Finally, we give some somewhat surprising examples of dual unital operator spaces.

Weak∗ continuous states on Banach algebras

“…To show that the set D A of all such elements is weak * closed, if A is a dual space, we will need a suitable metric characterization of dissipative elements (Lemma 2.1 below). A similar, but not the same, characterization was observed in [2] for C * -algebras; however, the argument from [2] does not apply to Banach algebras. For each a ∈ D A the element 1 − a is invertible since its numerical range (hence also its spectrum) is contained in the half-plane Re z 1.…”

“…In C * -algebras the estimate (2.1) can be improved to 1 + ta 2 1 + t 2 a 2 (a ∈ D A , t 0), a consequence of the C * -identity [2]. This sharper estimate holds also in some other natural examples of Banach algebras, but the author does not know if it holds in general.…”

“…The proof is the same as for operator spaces [2]. Since it is very short, we will sketch it here for completeness.…”

Metric characterizations of isometries and of unital operator spaces and systems

“…[6, Section 1.4]. The analogous fact for operator systems, or for unital operator spaces, was conjectured in an earlier draft of the present paper and later solved in [7] sa . Then (x s ) and (y s ) are bounded nets.…”

“…Some interesting new examples of unital operator spaces are presented in [7]. For example, the Fourier algebra B(G) of any amenable group, with its usual operator space structure as the dual of the group C * -algebra of G, is a unital operator space.…”