41.2 Operator Systems
Definition 41.2.1 (Concrete Operator System).label Let $A$ be a $C^{*}$-algebra and $S \subset A$, then $S$ is a concrete operator system if:
- (1)
$1_{A} \in S$.
- (2)
$S = S^{*}$.
- (3)
$S$ is a linear subspace of $A$.
Definition 41.2.2 (Self-Adjoint Part).label Let $A$ be a $C^{*}$-algebra and $S \subset A$ be an operator system, then $S_{sa}= \bracsn{x \in S|x = x^*}$ is the self-adjoint part of $S$.
Definition 41.2.3 (Positive Cone).label Let $A$ be a $C^{*}$-algebra and $S \subset A$ be an operator system, then $S_{+}= \bracsn{x \in S_{sa}|x \ge 0}$ is the positive cone of $S$.
Proposition 41.2.4.label Let $A$ be a $C^{*}$-algebra, $S \subset A$ be an operator system, and $x \in S_{sa}$, then there exists positive elements $p, q \in S_{sa}$ such that $x = p - q$.
Proof. Let $p = (\norm{x}_{A} + x)/2$ and $q = (\norm{x}_{A} - x)/2$, then $q$ is positive by Corollary 38.6.3.$\square$
Definition 41.2.5 (Positive).label Let $A, B$ be $C^{*}$-algebra, $S \subset A$ and $T \subset B$ be operator systems, and $\phi: S \to T$ be a linear map, then $\phi$ is positive if $\phi(S_{+}) \subset \phi(T_{+})$.
Proposition 41.2.6.label Let $A, B$ be $C^{*}$-algebra, $S \subset A$ and $T \subset B$ be operator systems, and $\phi: S \to T$ be a positive map, then
- (1)
For any $x \in S_{sa}$, $\norm{\phi(x)}_{B}\le \norm{\phi(1_A)}_{B} \cdot \norm{x}_{A}$.
- (2)
$\norm{\phi}_{L(S; T)}\le 2\norm{\phi(1_A)}_{B}$.
Proof. (1): Let $x \in T_{+}$, then $0 \le x \le \norm{x}_{A}$, and $0 \le \phi(x) \le \norm{x}_{A} \cdot \norm{\phi(1_A)}_{B}$, so $\norm{\phi(x)}_{B} \le \norm{x}_{A} \cdot \norm{\phi(1_A)}_{B}$.
For any $x, y \in T_{+}$, $x - y \le x$ and $y - x \le y$. As such,
In particular, for any $s \in S_{sa}$,
(2): For any general $s \in S$,
$\square$
Theorem 41.2.7.label Let $X$ be a compact Hausdorff space, $A$ be a unital $C^{*}$-algebra and $\phi: C(X; \complex) \to A$ be a positive map, then $\norm{\phi}_{L(C(X; \complex); B)}= \norm{\phi(\one)}_{B}$.
Proof. Assume without loss of generality that $\norm{\phi(\one)}_{B}\le 1$.
Let $\seqf{h_j}\subset C(X; [0, 1])$ be a partition of unity on $X$, and $\seqf{\lambda_j}\subset \ol{B_\complex(0, 1)}$.
Since $A$ is a unital $C^{*}$-algebra, there exists a complex Hilbert space $H$ such that $A$ may be identified with a $C^{*}$-subalgebra of $B(H)$, with $1_{A} = 1_{B(H)}$. In which case, for any $\xi, \eta \in H$,
so $\normn{\sum_{j = 1}^n \lambda_j\phi(h_j)}_{A} \le 1$.
Let $f \in C(X; \complex)$ with $\norm{f}_{u} \le 1$ and $\eps > 0$, then there exists a partition of unity $\seqf{h_j}\subset C(X; [0, 1])$ such that for each $1 \le j \le n$, $\sup_{x, y \in \supp{h_j}}|f(x) - f(y)| < \eps$. For each $1 \le j \le n$, choose $x_{j} \in \supp{h_j}$, then $\norm{f - \sum_{j = 1}^n f(x_j)h_j}_{u} \le \eps$.
By the previous case and Proposition 41.2.6,
$\square$
Example 41.2.8 (Arveson).label Let $\mathbb{T}= \bracsn{z \in \complex|\ |z| = 1}$, and $S = \text{span}\bracsn{1, z, \ol z}\subset C(\mathbb{T}; \complex)$, then $S$ is an operator system in $C(\mathbb{T}; \complex)$. Let
then:
- (1)
$\phi$ is a positive map.
- (2)
$\norm{\phi}_{L(C(\mathbb{T} ; \complex); M_2(\complex))}= 2$.
- (3)
$\phi$ admits no extension to a positive map on $C(\mathbb{T}; \complex)$.
As such, $M_{2}(\complex)$ is not injective with respect to operator systems and positive maps.
Proof, [Example 2.2, Po02]. (1): Let $a + bz + c \ol z \in S$, then the following are equivalent:
- (1)
$a + bz + c\ol z \ge 0$.
- (2)
$c = \ol b$ and $a \ge 2\text{Re}(b)$.
- (3)
$c = \ol b$ and $a \ge 2|b|$.
In addition, since a $2 \times 2$ matrix is positive if and only if its diagonal entries and its determinant are non-negative. Thus $\phi$ is positive.
(2): $\norm{\phi(z)}_{M_2(\complex)}= 2$.
(3): By Theorem 41.2.7, any such extension would have norm $1$.$\square$
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