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'state'
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| Title of object: |
state |
| Canonical Name: |
State |
| Type: |
Definition |
| Created on: |
2003-08-11 13:05:39 |
| Modified on: |
2004-03-07 15:55:30 |
| Classification: |
msc:46L05 |
| Defines: |
pure state, tracial state |
Preamble:
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Content:
A \defn{state} $\Psi$ on a $C^*$-algebra $A$ is a positive linear functional
$\Psi : A \to \Cset$, $\Psi(a^*a) \geq 0$ for all $a \in A$, with unit norm.
The norm of a positive linear functional is defined by
\begin{equation}
||\Psi|| = \sup_{a \in A}\{|\Psi(a)| : ||a||\leq 1\}.
\end{equation}
For a unital $C^*$-algebra, $||\Psi|| = \Psi(\identity)$.
The space of states is a convex set.
Let $\Psi_1$ and $\Psi_2$ be states, then the convex combination
\begin{equation}
\lambda\Psi_1+(1-\lambda)\Psi_2, \quad \lambda \in [0,1],
\end{equation}
is also a state.
A state is \defn{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 \defn{tracial state} if it is also a trace.
When a $C^*$-algebra is represented on a Hilbert space $\hilbert$,
every unit vector $\psi \in \hilbert$ determines a (not necessarily pure) state in the form of an \defn{expectation value},
\begin{equation}
\Psi(a) = \langle\psi, a\psi\rangle.
\end{equation}
In physics, it is common to refer to such states by their vector $\psi$ rather than the linear functional $\Psi$.
The converse is not always true; not every state need be given by
an 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).
\begin{thebibliography}{10}
\bibitem{Murphy}
G.~Murphy, {\em $C^*$-Algebras and Operator Theory}.
\newblock Academic Press, 1990.
\end{thebibliography} |
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