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Abstract

We study the issue of purity (as a completely positive linear map) for identity maps on
operator systems and for completely isometric embeddings of operator systems into their
C${}^{\ast}$ and injective
envelopes. Our most general result states that the canonical embedding of an operator system
$\mathcal{\mathcal{R}}$ into its injective
envelope
$I(\mathcal{\mathcal{R}})$ is pure if and
only if the C${}^{\ast}$envelope
${C}_{e}^{\ast}(\mathcal{\mathcal{R}})$ of
$\mathcal{\mathcal{R}}$ is a prime
C${}^{\ast}$algebra.
To prove this, we also show that the identity map on any
AW${}^{\ast}$factor
is a pure completely positive linear map.
For embeddings of operator systems
$\mathcal{\mathcal{R}}$ into their
C${}^{\ast}$envelopes,
the issue of purity is seemingly harder to describe in full generality,
so we focus here on operator systems arising from the generators of
discrete groups. Two such operator systems of interest are denoted by
${\mathcal{\mathcal{S}}}_{n}$ and
$\mathrm{NC}(n)$, where
${\mathcal{\mathcal{S}}}_{n}$ corresponds to the
generators of the free group
${\mathbb{\mathbb{F}}}_{n}$
and
$\mathrm{NC}(n)$ corresponds to the
generators of the group
${\mathbb{Z}}_{2}\ast \cdots \ast {\mathbb{Z}}_{2}$,
the free product of
$n$
copies of
${\mathbb{Z}}_{2}$. The
operator systems
${\mathcal{\mathcal{S}}}_{n}$
and
$\mathrm{NC}(n)$ are
of interest in operator theory for their connections to the weak expectation property and
C${}^{\ast}$nuclearity,
and for their universal properties. Specifically,
${\mathcal{\mathcal{S}}}_{n}$ is the universal operator
system for arbitrary
$n$tuples
of contractions acting on a Hilbert space and
$\mathrm{NC}(n)$ is the universal operator
system for
$n$tuples
of selfadjoint contractions. We show that the embedding of
${\mathcal{\mathcal{S}}}_{n}$ into
${C}_{e}^{\ast}\phantom{\rule{0.17em}{0ex}}(\mathcal{\mathcal{S}})$ is pure for all
$n\ge 2$ and that the
embedding of
$\mathrm{NC}(n)$
into
${C}_{e}^{\ast}(\mathrm{NC}(n))$ is pure
for every
$n\ge 3$.
The question of purity of the identity is quite subtle for operator systems that are not
C${}^{\ast}$algebras
and possibly must be handled on a casebycase basis. In this regard, we consider
the purity of the identity map on each of the universal operator systems
${\mathcal{\mathcal{S}}}_{n}$ and
$\mathrm{NC}(n)$.
Lastly, we present an unrecorded feature of pure completely positive linear maps,
namely that every pure completely positive linear map from an operator system
$\mathcal{\mathcal{R}}$ into an
injective factor
$\mathcal{\mathcal{M}}$
has a pure completely positive extension to any operator system
$\mathcal{\mathcal{T}}$ that
contains
$\mathcal{\mathcal{R}}$
(as an operator subsystem), thus generalizing a result of Arveson for the injective type I
factor $\mathcal{\mathcal{B}}(\mathcal{\mathscr{H}})$.

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Keywords
operator system, pure completely positive linear map,
C$^*$envelope, prime C$^*$algebra, AW$^*$factor,
injective envelope

Mathematical Subject Classification 2010
Primary: 46L07, 47L05, 47L07
Secondary: 46A32, 46A55, 47L25

Milestones
Received: 23 January 2020
Revised: 21 September 2020
Accepted: 23 April 2022
Published: 14 July 2022

