Purity of the embeddings of operator systems into their C$^*$- and injective envelopes

Douglas Farenick and Ryan Tessier

Vol. 317 (2022), No. 2, 317–338

DOI: 10.2140/pjm.2022.317.317

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 $\mathcal{ℛ}$ into its injective envelope $I<!--FUNCTION\; APPLICATION-->(\mathcal{ℛ})$ is pure if and only if the C${}^{\ast }$-envelope ${C<!--FUNCTION\; APPLICATION-->}_{e<!--FUNCTION\; APPLICATION-->}^{\ast }(\mathcal{ℛ})$ of $\mathcal{ℛ}$ 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 $\mathcal{ℛ}$ 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 ${\mathcal{𝒮}}_n$ and $\mathrm{NC}<!--FUNCTION\; APPLICATION-->(n)$, where ${\mathcal{𝒮}}_n$ corresponds to the generators of the free group ${\mathbb{𝔽}}_n$ and $\mathrm{NC}<!--FUNCTION\; APPLICATION-->(n)$ 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 ${\mathcal{𝒮}}_n$ and $\mathrm{NC}<!--FUNCTION\; APPLICATION-->(n)$ are of interest in operator theory for their connections to the weak expectation property and C${}^{\ast }$-nuclearity, and for their universal properties. Specifically, ${\mathcal{𝒮}}_n$ is the universal operator system for arbitrary $n$-tuples of contractions acting on a Hilbert space and $\mathrm{NC}<!--FUNCTION\; APPLICATION-->(n)$ is the universal operator system for $n$-tuples of selfadjoint contractions. We show that the embedding of ${\mathcal{𝒮}}_n$ into ${C<!--FUNCTION\; APPLICATION-->}_{e<!--FUNCTION\; APPLICATION-->}^{\ast }(\mathcal{𝒮})$ is pure for all $n\ge 2$ and that the embedding of $\mathrm{NC}<!--FUNCTION\; APPLICATION-->(n)$ into ${C<!--FUNCTION\; APPLICATION-->}_{e<!--FUNCTION\; APPLICATION-->}^{\ast }(\mathrm{NC}<!--FUNCTION\; APPLICATION-->(n))$ 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 case-by-case basis. In this regard, we consider the purity of the identity map on each of the universal operator systems ${\mathcal{𝒮}}_n$ and $\mathrm{NC}<!--FUNCTION\; APPLICATION-->(n)$.

Lastly, we present an unrecorded feature of pure completely positive linear maps, namely that every pure completely positive linear map from an operator system $\mathcal{ℛ}$ into an injective factor $\mathcal{ℳ}$ has a pure completely positive extension to any operator system $\mathcal{𝒯}$ that contains $\mathcal{ℛ}$ (as an operator subsystem), thus generalizing a result of Arveson for the injective type I factor $\mathcal{ℬ}(\mathcal{\mathscr{H}})$.

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