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Version 2 |
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A \textbf{state} $\Psi$ on a $C^*$-algebra $A$ is a positive linear functional
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A \defn{state} $\Psi$ on a $C^*$-algebra $A$ is a positive linear functional
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$\Psi\colon A \to \Cset$, $\Psi(a^*a) \geq 0$ for all $a \in A$, with unit norm.
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$\Psi : A \to \Cset$, $\Psi(a^*a) \geq 0$ for all $a \in A$, with unit norm.
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| The norm of a positive linear functional is defined by |
The norm of a positive linear functional is defined by |
| \begin{equation} |
\begin{equation} |
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\norm{\Psi} = \sup_{a \in A}\{|\Psi(a)| : \norm{a}\leq 1\}.
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||\Psi|| = \sup_{a \in A}\{|\Psi(a)| : ||a||\leq 1\}.
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| \end{equation} |
\end{equation} |
| For a unital $C^*$-algebra, $\norm{\Psi} = \Psi(\identity)$. |
For a unital $C^*$-algebra, $||\Psi|| = \Psi(\identity)$. |
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| The space of states is a convex set. |
The space of states is a convex set. |
| Let $\Psi_1$ and $\Psi_2$ be states, then the convex combination |
Let $\Psi_1$ and $\Psi_2$ be states, then the convex combination |
| \begin{equation} |
\begin{equation} |
| \lambda\Psi_1+(1-\lambda)\Psi_2, \quad \lambda \in [0,1], |
\lambda\Psi_1+(1-\lambda)\Psi_2, \quad \lambda \in [0,1], |
| \end{equation} |
\end{equation} |
| is also a state. |
is also a state. |
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A state is \defn{pure} if it is not a convex combination of two other states. |
| A state is \textbf{pure} if it is not a convex combination of two other states. |
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| Pure states are the extreme points of the convex set of 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 pure state on a commutative $C^*$-algebra is equivalent to a character. |
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A state is called a \defn{tracial state} if it is also a trace. |
| A state is called a \textbf{tracial state} if it is also a trace. |
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| When a $C^*$-algebra is represented on a Hilbert space $\hilbert$, |
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}, |
every unit vector $\psi \in \hilbert$ determines a (not necessarily pure) state in the form of an \defn{expectation value}, |
| \begin{equation} |
\begin{equation} |
| \Psi(a) = \langle\psi, a\psi\rangle. |
\Psi(a) = \langle\psi, a\psi\rangle. |
| \end{equation} |
\end{equation} |
| In physics, it is common to refer to such states by their vector $\psi$ rather than the linear functional $\Psi$. |
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 |
The converse is not always true; not every state need be given by |
| an expectation value. |
an expectation value. |
| For example, delta functions (which are distributions not functions) |
For example, delta functions (which are distributions not functions) |
| give pure states on $C_0(X)$, |
give pure states on $C_0(X)$, |
| but they do not correspond to any vector in a Hilbert space |
but they do not correspond to any vector in a Hilbert space |
| (such a vector would not be square-integrable). |
(such a vector would not be square-integrable). |
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|
| \begin{thebibliography}{10} |
\begin{thebibliography}{10} |
| \bibitem{Murphy} |
\bibitem{Murphy} |
| G.~Murphy, {\em $C^*$-Algebras and Operator Theory}. |
G.~Murphy, {\em $C^*$-Algebras and Operator Theory}. |
| \newblock Academic Press, 1990. |
\newblock Academic Press, 1990. |
| \end{thebibliography} |
\end{thebibliography} |
