state

A state $\mathrm{\Psi }$ on a $C^{*}$-algebra $A$ is a positive linear functional$\mathrm{\Psi }:A\to ℂ$, $\mathrm{\Psi }(a^{*}a)\ge 0$ for all $a\in A$, with unit norm.The norm of a positive linear functional is defined by

$$\parallel \mathrm{\Psi }\parallel =\underset{a\in A}{sup}\{|\mathrm{\Psi }(a)|:\parallel a\parallel \le 1\}.$$(1)

For a unital $C^{*}$-algebra, $\parallel \mathrm{\Psi }\parallel =\mathrm{\Psi }(1\mathrm{I})$.

The space of states is a convex set.Let ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ be states, then the convex combination

$$\lambda {\mathrm{\Psi }}_1+(1-\lambda ){\mathrm{\Psi }}_2,\lambda \in [0,1],$$(2)

is also a state.

A state is pure if it is not a convex combination of two other states.Pure states are the extreme points of the convex set of states.A pure state on a commutative $C^{*}$-algebra is equivalent to a character.

A state is called a tracial state if it is also a trace.

When a $C^{*}$-algebra is represented on a Hilbert space $\mathscr{H}$,every unit vector $\psi \in \mathscr{H}$ determines a (not necessarily pure) state in the form of an expectation value,

$$\mathrm{\Psi }(a)=⟨\psi ,a\psi ⟩.$$(3)

In physics, it is common to refer to such states by their vector $\psi$ rather than the linear functional $\mathrm{\Psi }$.The converse is not always true; not every state need be given byan expectation value.For example, delta functions (which are distributions not functions)give pure states on $C_0(X)$,but they do not correspond to any vector in a Hilbert space(such a vector would not be square-integrable).