# 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 $\phi(x)=0$ for all $\phi\in S(\mathcal{A})$.

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

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

• If $x$ is normal (http://planetmath.org/InvolutaryRing), then $\phi(x)=\|x\|$ for some $\phi\in S(\mathcal{A})$.

## 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 $\phi(x)=0$ for all $\phi\in P(\mathcal{A})$.

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

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

• If $x$ is , then $\phi(x)=\|x\|$ for some $\phi\in P(\mathcal{A})$.

- Every multiplicative linear functional on $\mathcal{A}$ is a pure state.

Title properties of states PropertiesOfStates 2013-03-22 17:45:24 2013-03-22 17:45:24 asteroid (17536) asteroid (17536) 5 asteroid (17536) Theorem msc 46L30 msc 46L05