state space is non-empty

In this entry we prove the existence of states for every $C^{*}$ -algebra (http://planetmath.org/CALgebra).

Theorem - Let $\mathcal{A}$ be a $C^{*}$ -algebra. For every self-adjoint (http://planetmath.org/InvolutaryRing) element $a\in\mathcal{A}$ there exists a state $\psi$ on $\mathcal{A}$ such that $|\psi(a)|=\|a\|$ .

$\;$

Proof : We first consider the case where $\mathcal{A}$ is unital (http://planetmath.org/Ring), with identity element $e$ .

Let $\mathcal{B}$ be the $C^{*}$ -subalgebra generated by $a$ and $e$ . Since $a$ is self-adjoint, $\mathcal{B}$ is a comutative $C^{*}$ -algebra with identity element.

Thus, by the Gelfand-Naimark theorem, $\mathcal{B}$ is isomorphic to $C(X)$ , the space of continuous functions $X\longrightarrow\mathbb{C}$ for some compact set $X$ .

Regarding $a$ as an element of $C(X)$ , $a$ attains a maximum at a point $x_{0}\in X$ , since $X$ is compact. Hence, $\|a\|=|a(x_{0})|$ .

The evaluation function at $x_{0}$ ,

	$\displaystyle ev_{x_{0}}:C(X)\longrightarrow\mathbb{C}$
	$\displaystyle ev_{x_{0}}(f):=f(x_{0})$

is a multiplicative linear functional of $C(X)$ . Hence, $\|ev_{x_{0}}\|=1$ and also $|ev_{x_{0}}(a)|=|a(x_{0})|=\|a\|$ .

We can now extend $ev_{x_{0}}$ to a linear functional $\psi$ on $\mathcal{A}$ such that $\|\psi\|=\|ev_{x_{0}}\|=1$ , using the Hahn-Banach theorem.

Also, $\psi(e)=ev_{x_{0}}(e)=1$ and so $\psi$ is a norm one positive linear functional, i.e. $\psi$ is a state on $\mathcal{A}$ .

Of course, $\psi$ is such that $|\psi(a)|=|ev_{x_{0}}(a)|=\|a\|$ .

In case $\mathcal{A}$ does not have an identity element we can consider its minimal unitization $\widetilde{\mathcal{A}}$ . By the preceding there is a state $\widetilde{\psi}$ on $\widetilde{\mathcal{A}}$ satisfying the required . Now, we just need to take the restriction (http://planetmath.org/RestrictionOfAFunction) of $\widetilde{\psi}$ to $\mathcal{A}$ and this restriction is a state in $\mathcal{A}$ satisfying the required . $\square$

Title	state space is non-empty
Canonical name	StateSpaceIsNonempty
Date of creation	2013-03-22 17:45:14
Last modified on	2013-03-22 17:45:14
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	14
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 46L30
Classification	msc 46L05