Corollary 34.10.9.label Let $A$ be a unital $C^{*}$-algebra, then:
- (1)
For each $x \in A$, $x = 0$ if and only if $\dpn{x, \phi}{A}= 0$ for all $\phi \in P(A)$.
- (2)
The linear span of $P(A)$ is weak*-dense in $A^{*}$.
Moreover, for any $x \in A$,
- (3)
$x$ is self-adjoint if and only if $\dpn{x, \phi}{A}\in \real$ for all $\phi \in P(A)$.
- (4)
$x$ is positive if and only if $\dpn{x, \phi}{A}\ge 0$ for all $\phi \in P(A)$.
Proof, [Theorem 13.9, Zhu93]. (1): Let $x \in A$ such that $\dpn{x, \phi}{A}= 0$ for all $\phi \in P(A)$. First suppose that $x$ is self-adjoint. By Theorem 34.10.8, $\sigma_{A}(x) = \bracs{0}$, and $\norm{x}_{A} = [x]_{sp}= 0$ by Theorem 34.4.3.
Now suppose that $x$ is arbitrary. In this case, for each $\phi \in P(A)$,
because $\phi$ is Hermitian. Similarly, $\dpn{\text{Im}(x), \phi}{A}= 0$ as well. Thus $\text{Re}(x) = \text{Im}(x) = 0$, and $x = 0$ as well.
(2): Since the linear span of $P(A)$ separates points in $A$, it is weak*-dense in $A^{*}$ by Lemma 17.1.5.
(3): Let $\phi \in P(A)$, then $\phi$ is Hermitian. If $x$ is self-adjoint, then $\dpn{x, \phi}{A}\in \real$.
On the other hand, if $\dpn{x, \phi}{A}\in \real$, then $\dpn{x, \phi}{A}= \dpn{x^*, \phi}{A}$, and $\dpn{x - x^*, \phi}{A}=0$. If this holds for all $\phi \in P(A)$, then $x - x^{*} = 0$ by (1), and $x$ is self-adjoint.
(4): Let $\phi \in P(A)$, then $\phi$ is positive. Thus if $x$ is positive, $\dpn{x, \phi}{A}\ge 0$.
On the other hand, if $\dpn{x, \phi}{A}\ge 0$ for all $\phi \in P(A)$, then $x$ is self-adjoint by (3). By Corollary 34.10.7, $\sigma_{A}(x) \subset [0, \infty)$. As such, $x$ is positive by Corollary 34.6.3.$\square$
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