properties of states
Let $\mathcal{A}$ be a ${C}^{*}$algebra (http://planetmath.org/CAlgebra) and $x\in \mathcal{A}$.
Let $S(\mathcal{A})$ and $P(\mathcal{A})$ denote the state (http://planetmath.org/State) space and the pure state space of $\mathcal{A}$, respectively.
0.1 States
The space is sufficiently large to reveal many of elements of a ${C}^{*}$algebra.
Theorem 1 We have that

•
$S(\mathcal{A})$ separates points, i.e. $x=0$ if and only if $\varphi (x)=0$ for all $\varphi \in S(\mathcal{A})$.

•
$x$ is selfadjoint^{} (http://planetmath.org/InvolutaryRing) if and only if $\varphi (x)\in \mathbb{R}$ for all $\varphi \in S(\mathcal{A})$.

•
$x$ is positive if and only if $\varphi (x)\ge 0$ for all $\varphi \in S(\mathcal{A})$.

•
If $x$ is normal (http://planetmath.org/InvolutaryRing), then $\varphi (x)=\parallel x\parallel $ for some $\varphi \in S(\mathcal{A})$.
0.2 Pure states
The pure state space is also sufficiently large to the of Theorem 1. Hence, we can replace $S(\mathcal{A})$ by $P(\mathcal{A})$, or by any other family of linear functionals^{} $F$ such that $P(\mathcal{A})\subset F\subset S(\mathcal{A})$, in the previous result.
Theorem 2  We have that

•
$P(\mathcal{A})$ separates points, i.e. $x=0$ if and only if $\varphi (x)=0$ for all $\varphi \in P(\mathcal{A})$.

•
$x$ is if and only if $\varphi (x)\in \mathbb{R}$ for all $\varphi \in P(\mathcal{A})$.

•
$x$ is positive if and only if $\varphi (x)\ge 0$ for all $\varphi \in P(\mathcal{A})$.

•
If $x$ is , then $\varphi (x)=\parallel x\parallel $ for some $\varphi \in P(\mathcal{A})$.
 Every multiplicative linear functional on $\mathcal{A}$ is a pure state.
Title  properties of states 

Canonical name  PropertiesOfStates 
Date of creation  20130322 17:45:24 
Last modified on  20130322 17:45:24 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  5 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 46L30 
Classification  msc 46L05 