properties of states

Let $𝒜$ be a $C^{*}$-algebra (http://planetmath.org/CAlgebra) and $x\in 𝒜$.

Let $S(𝒜)$ and $P(𝒜)$ denote the state (http://planetmath.org/State) space and the pure state space of $𝒜$, respectively.

0.1 States

The space is sufficiently large to reveal many of elements of a $C^{*}$-algebra.

Theorem 1- We have that

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

• $x$ is self-adjoint (http://planetmath.org/InvolutaryRing) if and only if $\varphi (x)\in ℝ$ for all $\varphi \in S(𝒜)$.

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

• If $x$ is normal (http://planetmath.org/InvolutaryRing), then $\varphi (x)=\parallel x\parallel$ for some $\varphi \in S(𝒜)$.

0.2 Pure states

The pure state space is also sufficiently large to the of Theorem 1. Hence, we can replace $S(𝒜)$ by $P(𝒜)$, or by any other family of linear functionals $F$ such that $P(𝒜)\subset F\subset S(𝒜)$, in the previous result.

Theorem 2 - We have that

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

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

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

• If $x$ is , then $\varphi (x)=\parallel x\parallel$ for some $\varphi \in P(𝒜)$.

- Every multiplicative linear functional on $𝒜$ is a pure state.