Duality for operator systems with generating cones
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ID: 282303
2025
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Abstract
Let $S$ be a complete operator system with a generating cone; i.e. $S_\sa =
S_+ - S_+$. We show that there is a matrix norm on the dual space $S^*$, under
which, and the usual dual matrix cone, $S^*$ becomes a dual operator system
with a generating cone, denoted by $S^\rd$. The canonical complete order
isomorphism $\iota_{S^*}: S^* \to S^\rd$ is a dual Banach space isomorphism.
Furthermore, we construct a canonical completely contractive
weak$^*$-homeomorphism $\beta_S: (S^\rd)^\rd\to S^{**}$, and verify that it is
a complete order isomorphism.
For a complete operator system $T$ with a generating cone and a completely
positive complete contraction $\varphi:S\to T$, there is a weak$^*$-continuous
completely positive complete contraction $\varphi^\rd:T^\rd \to S^\rd$ with
$\iota_{S^*}\circ \varphi^* = \varphi^\rd \circ \iota_{T^*}$. This produces a
faithful functor from the category of complete operator systems with generating
cones (where morphisms are completely positive complete contractions) to the
category of dual operator systems with generating cones (where morphisms are
weak$^*$-continuous completely positive complete contractions).
We define the notion of approximately unital operator systems, and verify
that operator systems considered in \cite{CvS} and \cite{CvS2} are
approximately unital. If $S$ is approximately unital, then $\iota_{S^*}:S^* \to
S^\rd$ is an operator space isomorphism and $\beta_S: (S^\rd)^\rd\to S^{**}$ is
a complete isometry. We will also establish that the restriction of the
faithful functor $(S,T,\varphi)\mapsto (T^\rd, S^\rd, \varphi^\rd)$ to the
category of approximately unital complete operator systems is both full and
injective on objects.
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| Authors | Yu-Shu Jia; Chi-Keung Ng |
| Journal | arXiv |
| Year | 2025 |
| DOI |
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