\section{Injective Operator Systems}
\label{section:injective-ops}


By \hyperref[Arveson's Extension Theorem]{theorem:arveson-extension}, $B(H)$ is injective for any Hilbert space $H$. Further, the \hyperref[Choi-Effros Theorem]{theorem:choi-effros} implies that every operator system embeds into $B(H)$, for a suitable Hilbert space $H$. Therefore the category of operator systems, with unital complete order isomorphisms, has sufficiently many injectives. 

\begin{theorem}
\label{theorem:opsys-predual}
    Let $S, T$ be operator systems. If $T$ admits a predual, then the space
    \[
    UCP(S; T) = \bracsn{\phi: S \to T| \phi \text{ unital, completely positive}}
    \]

    is compact in the point-weak* topology. 
\end{theorem}


\begin{definition}[Injective]
\label{definition:injective-operator-system}
    Let $I$ be an operator system, then $I$ is \textbf{injective} if for any pair of operator systems $S, T$ with $S \subset T$ and unital completely positive map $\phi: S \to I$, there exists a unital completely positive map $\Phi: T \to I$ such that the following diagram commutes
    \[
    \xymatrix{
T \ar@{->}[r]^{\Phi} & I \\
S \ar@{->}[ru]_{\phi} \ar@{^{(}->}[u] & 
}
    \]
\end{definition}

\begin{definition}[Conditional Expectation]
\label{definition:os-conditional-expectation}
    Let $S, T$ be operator systems with $S \subset T$ and $\phi: T \to S$ be a unital completely positive map, then $\phi$ is a \textbf{conditional expectation} if $\phi|_{T} = \text{Id}_T$. 
\end{definition}


\begin{proposition}
\label{proposition:os-injective-ce}
    Let $I$ be an operator system, then the following are equivalent:
    \begin{enumerate}
        \item $I$ is injective. 
        \item For every complex Hilbert space $H$ and embedding $\iota: I \to B(H)$, there exists a conditional expectation $\phi: B(H) \to \iota(I)$. In other words, $\phi$ is a unital completely positive map such that the following diagram commutes
        \[
        \xymatrix{
        B(H) \ar@{->}[r]^{\phi} & \iota(I) \\
        I \ar@{=}[ru] \ar@{^{(}->}[u] & 
        }
        \]
    \end{enumerate}
\end{proposition}
\begin{proof}
    (2) $\Rightarrow$ (1): By the \hyperref[Choi-Effros Theorem]{theorem:choi-effros} and \hyperref[Arveson's Extension Theorem]{theorem:arveson-extension}. 
\end{proof}
% "obvious"

\begin{example}[Dixmier Algebra]
    \label{example:sup-not-vna}
    Let
    \[
    A = B^\infty([0, 1])/\bracsn{f \in B^\infty([0, 1])| \bracsn{f \ne 0} \text{ is meagre}}
    \]
    
    then $A$ is an injective, order complete $C^*$-algebra, but $A$ is not a von Neumann algebra. 
\end{example}


\begin{proposition}
\label{proposition:os-injective-cstar}
    Let $H$ be a complex Hilbert space, $I \subset B(H)$ be an operator system, and $\phi: B(H) \to I$ be a conditional expectation, then $(x, y) \mapsto \phi(xy)$ is a well-defined product on $I$, making it a $C^*$-algebra. 
\end{proposition}


\begin{proposition}
\label{proposition:os-injective-cstar}
    Let $I$ be an injective operator system, then there exists a $C^*$-algebra $A$ such that $I$ is isomorphic to $A$ as operator systems. 
\end{proposition}

\begin{proposition}
\label{proposition:os-maximal-essential-extension}
    Let $S$ be an operator system, then there exists a pair $(\wh S, \iota)$ such that:
    \begin{enumerate}
        \item $(\wh S, \iota)$ is an essential extension of $S$. 
        \item For any proper\footnote{Non-isomorphic.} extension $(T, j)$ of $\wh S$, $(Z, j \circ \iota)$ is not an essential extension of $S$. In other words, every essential extension of $(\wh S, \iota)$ is isomorphic to $(\wh S, \iota)$.
    \end{enumerate} 

    The pair $(\wh S, \iota)$ is a \textbf{maximal essential extension} of $S$. 
\end{proposition}
\begin{proof}
    Zorn's lemma. 
\end{proof}

\begin{theorem}
\label{theorem:maximal-essential-injective}
    
\end{theorem}



% I sure wonder how that will happen