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Diffstat (limited to 'pages/operator-algebras/c-star-algebras/states.md')
| -rw-r--r-- | pages/operator-algebras/c-star-algebras/states.md | 30 |
1 files changed, 18 insertions, 12 deletions
diff --git a/pages/operator-algebras/c-star-algebras/states.md b/pages/operator-algebras/c-star-algebras/states.md index 619bc9a..29cf5f5 100644 --- a/pages/operator-algebras/c-star-algebras/states.md +++ b/pages/operator-algebras/c-star-algebras/states.md @@ -3,27 +3,28 @@ title: States parent: C*-Algebras grand_parent: Operator Algebras nav_order: 1 -# cspell:words --- # {{ page.title }} -{: .definition-title } -> Definition (State, State Space) -> -> A norm-one [positive linear functional]( {% link pages/operator-algebras/c-star-algebras/positive-linear-functionals.md %} ) on a C\*-algebra is called a *state*.\ -> The *state space* $S(\mathcal{A})$ of a C\*-algebra $\mathcal{A}$ is the set of all its states. +{% definition State, State Space %} +A norm-one [positive linear functional]( {% link pages/operator-algebras/c-star-algebras/positive-linear-functionals.md %} ) on a C\*-algebra is called a *state*.\ +The *state space* $S(\mathcal{A})$ of a C\*-algebra $\mathcal{A}$ is the set of all its states. +{% enddefinition %} -Note that $S(\mathcal{A})$ is a subset of the unit ball in the dual space of $\mathcal{A}$. +Note that $S(\mathcal{A})$ is a subset of the closed unit ball in the dual space of $\mathcal{A}$. -{: .corollary } -> A linear functional $\omega$ on a C\*-algebra is a state -> if and only if $\omega(\mathbf{1}) = 1 = \norm{\omega}$. +{% corollary %} +A linear functional $\omega$ on a C\*-algebra is a state +if and only if $\omega(\mathbf{1}) = 1 = \norm{\omega}$. +{% endcorollary %} -{: .proposition } -> The state space of a C\*-algebra is convex and weak\* compact. +{% proposition %} +The state space of a C\*-algebra is convex and weak\* compact. +{% endproposition %} {% proof %} +Let $\mathcal{A}$ be a C\*-algebra and let $S(\mathcal{A})$ be its state space. First, we show convexity. Let $\omega_0, \omega_1$ be states on $\mathcal{A}$ and let $t \in (0,1)$. Consider the convex combination $\omega = (1-t)\omega_0 + t\omega_1$. @@ -41,3 +42,8 @@ This means that $\omega_i(x) \to \omega(x)$ for every $x \in \mathcal{A}$. For all $i$ we have $\omega_i(x) \ge 0$ for $x \ge 0$ and $\omega_i(\mathbf{1}) = 1$; hence $\omega(x) \ge 0$ for $x \ge 0$ and $\omega(\mathbf{1}) = 1$. Thus $\omega$ is again a state. This shows that the state space is weak* closed, completing the proof. {% endproof %} + +TODO: state space is nonempty + +TODO: pure states + |