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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/operator-algebras/c-star-algebras | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
Update
Diffstat (limited to 'pages/operator-algebras/c-star-algebras')
-rw-r--r-- | pages/operator-algebras/c-star-algebras/positive-linear-functionals.md | 40 | ||||
-rw-r--r-- | pages/operator-algebras/c-star-algebras/states.md | 30 |
2 files changed, 36 insertions, 34 deletions
diff --git a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md index 05b1d4f..ea15f87 100644 --- a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md +++ b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md @@ -3,37 +3,33 @@ title: Positive Linear Functionals parent: C*-Algebras grand_parent: Operator Algebras nav_order: 1 -# cspell:words --- # {{ page.title }} all algebra are assumed to be unital -{: .definition-title } -> Hermitian Functional, Positive Functional, State -> -> A linear functional on a $C^*$-algebra $\mathcal{A}$ is said to be -> -> - *Hermitian* if $\phi(x*) = \overline{\phi(x)}$ for all $x \in \mathcal{A}$. -> - *positive* if $\phi(x) \ge 0$ for all $x \ge 0$. -> - a *state* if $\phi$ is positive and $\phi(\mathbf{1}) = 1$. -> - -{: .definition-title } -> State -> -> A norm-one positive linear functional on a $C^*$-algebra is called a *state*. - -{: .definition-title } -> State Space -> -> The *state space* of a $C^*$-algebra $\mathcal{A}$, denoted by $S(\mathcal{A})$, is the set of all states of $\mathcal{A}$. +{% definition Hermitian Functional, Positive Functional, State %} +A linear functional on a $C^*$-algebra $\mathcal{A}$ is said to be + +- *Hermitian* if $\phi(x*) = \overline{\phi(x)}$ for all $x \in \mathcal{A}$. +- *positive* if $\phi(x) \ge 0$ for all $x \ge 0$. +- a *state* if $\phi$ is positive and $\phi(\mathbf{1}) = 1$. +{% enddefinition %} + +{% definition State %} +A norm-one positive linear functional on a $C^*$-algebra is called a *state*. +{% enddefinition %} + +{% definition State Space %} +The *state space* of a $C^*$-algebra $\mathcal{A}$, denoted by $S(\mathcal{A})$, is the set of all states of $\mathcal{A}$. +{% enddefinition %} Note that $S(\mathcal{A})$ is a subset of the unit ball in the dual space of $\mathcal{A}$. -{: .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 %} {% endproof %} 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 + |