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Diffstat (limited to 'pages/operator-algebras')
4 files changed, 104 insertions, 93 deletions
diff --git a/pages/operator-algebras/banach-algebras/index.md b/pages/operator-algebras/banach-algebras/index.md index 2fb8f03..3335d78 100644 --- a/pages/operator-algebras/banach-algebras/index.md +++ b/pages/operator-algebras/banach-algebras/index.md @@ -47,25 +47,23 @@ We say that $\mathcal{A}$ is an *unital* Banach algebra, if $\mathcal{A}$ contai It is easy to see that a Banach algebra has at most one unit. -{: .proposition-title #neumann-series } -> Proposition (Neumann Series) -> -> Let $\mathcal{A}$ be a unital Banach algebra -> and let $x \in \mathcal{A}$ satisfy $\norm{x} < 1$. -> Then $\mathbf{1}-x$ is invertible -> and the inverse is given by the series -> -> $$ -> (\mathbf{1}-x)^{-1} = \sum_{n=0}^{\infty} x^n, -> $$ -> -> which converges absolutely in norm. -> Moreover, we have the estimate -> -> $$ -> \norm{(\mathbf{1}-x)^{-1}} \le \frac{1}{1 - \norm{x}}. -> $$ -> {: .katex-display .mb-0 } +{% proposition Neumann Series %} +Let $\mathcal{A}$ be a unital Banach algebra +and let $x \in \mathcal{A}$ satisfy $\norm{x} < 1$. +Then $\mathbf{1}-x$ is invertible, +and the inverse is given by the series + +$$ +(\mathbf{1}-x)^{-1} = \sum_{n=0}^{\infty} x^n, +$$ + +which converges absolutely in norm. +Moreover, we have the estimate + +$$ +\norm{(\mathbf{1}-x)^{-1}} \le \frac{1}{1 - \norm{x}}. +$$ +{% endproposition %} {% proof %} Since the Banach algebra norm is submultiplicative, @@ -91,18 +89,17 @@ The estimate follows from $\norm{s} \le \sum \norm{x}^n = 1 / (1 - \norm{x})$. ## The Spectrum -{: .definition-title } -> Definition (Spectrum, Resolvent Set) -> -> Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$. -> {: .mb-0 } -> -> {: .my-0 } -> - The *spectrum* of $x$ is the set $\sigma(x) = \lbrace\lambda \in \CC : x - \lambda$ is not invertible in $\mathcal{A}\rbrace$. \ -> The elements of $\sigma(x)$ are called *spectral values* of $x$. -> - The *resolvent set* of $x$ is the set $\rho (x) = \CC \setminus \sigma(x)$. \ -> For $\lambda \in \rho(x)$ the *resolvent* of $x$ is the algebra element $R_{\lambda} = (\lambda - x)^{-1}$. \ -> The mapping $R : \rho(x) \to \mathcal{A}$, $\lambda \mapsto R_{\lambda}$, is called *resolvent map*. +{% definition Spectrum, Resolvent Set %} +Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$. +{: .mb-0 } + +{: .my-0 } +- The *spectrum* of $x$ is the set $\sigma(x) = \lbrace\lambda \in \CC : x - \lambda$ is not invertible in $\mathcal{A}\rbrace$. \ + The elements of $\sigma(x)$ are called *spectral values* of $x$. +- The *resolvent set* of $x$ is the set $\rho (x) = \CC \setminus \sigma(x)$. \ + For $\lambda \in \rho(x)$ the *resolvent* of $x$ is the algebra element $R_{\lambda} = (\lambda - x)^{-1}$. \ + The mapping $R : \rho(x) \to \mathcal{A}$, $\lambda \mapsto R_{\lambda}$, is called *resolvent map*. +{% enddefinition %} {% theorem %} Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$. @@ -122,7 +119,7 @@ $$ {% proof %} Let $\lambda$ be in the resolvent set of $x$. -Then $\lambda - x$ is invertible and we have for all $\mu \in \CC$ +Then $\lambda - x$ is invertible, and we have for all $\mu \in \CC$ $$ \mu - x = \bigl(\mathbf{1} - (\lambda - \mu) (\lambda - x)^{-1}\bigr) (\lambda - x). @@ -165,7 +162,7 @@ We assume that $\sigma(x)$ is empty and derive a contradiction. Observe that the resolvent map $R$ is defined on the whole complex plane. By [this corollary](#resolvent-map-is-analytic), $R$ is analytic, hence entire. -Analytic functions are countinuous; +Analytic functions are continuous; therefore $R$ is bounded on the compact disk $\abs{\lambda} \le 2 \norm{x}$. For $\abs{\lambda} > 2 \norm{x}$ we may expand $R_{\lambda}$ into a [Neumann series](#neumann-series), @@ -188,13 +185,12 @@ $$ This shows that $R$ is a bounded entire function. Now [Liouville's Theorem](/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.html#liouvilles-theorem) (for vector-valued functions) implies that $R$ is constant. -This is contradictiory because XXX +This is contradictory because XXX {% endproof %} -{: .theorem-title } -> Gelfand–Mazur Theorem -> -> Every Banach algebra in which all nonzero elements are invertible is isometrically isomorphic to $\CC$. +{% theorem * Gelfand–Mazur Theorem %} +Every Banach algebra in which all nonzero elements are invertible is isometrically isomorphic to $\CC$. +{% endtheorem %} {% proof %} For any Banach algebra $A$, @@ -220,40 +216,39 @@ include: - $\mathcal{A}$ is a division algebra. - The underlying ring of $\mathcal{A}$ is a field. -{: .theorem-title } -> Spectral Radius Formula -> -> For every Banach algebra element $x$ the spectral radius is given by -> -> $$ -> r(x) = \lim_{n \to \infty} \norm{x^n}^{1/n}. -> $$ -> {: .katex-display .mb-0 } +{% theorem * Spectral Radius Formula %} +For every Banach algebra element $x$ the spectral radius is given by + +$$ +r(x) = \lim_{n \to \infty} \norm{x^n}^{1/n}. +$$ +{% endtheorem %} ## Gelfand’s Theory -Proposition +{% proposition %} Let $\mathcal{A}$ be a unital commutative Banach algebra. If $\phi$ is a nonzero multiplicative linear functional on $\mathcal{A}$, then its kernel $\ker \phi$ is a maximal ideal in $\mathcal{A}$. Every maximal ideal $\mathcal{I}$ in $\mathcal{A}$ is of the form $I = \ker \phi$ for some nonzero multiplicative linear functional $\phi$ on $\mathcal{A}$. +{% endproposition %} -In other words, the mapping $\phi \mapsto \ker \phi$ is gives a bijection +In other words, the mapping $\phi \mapsto \ker \phi$ gives a bijection between the sets of nonzero multiplicative linear functionals and maximal ideals. +{% definition %} +The *Gelfand space* $\Gamma_{\mathcal{A}}$ of a unital commutative Banach algebra $\mathcal{A}$ +is the set of maximal ideals of $\mathcal{A}$; its topology is inherited from +the weak* topology on the dual of $\mathcal{A}$ via the correspondence described above. +{% enddefinition %} -Definition +{% definition %} The *maximal ideal space* $\mathcal{M}_{\mathcal{A}}$ of a unital commutative Banach algebra $\mathcal{A}$ is the set of maximal ideals of $\mathcal{A}$; its topology is inherited from -the weak* topology on the dual of $\mathcal{A}$ via the correspondece described above. - -Proposition -The *maximal ideal space* of a unital commutative Banach algebra is a compact Hausdorff space. - -{% definition bla, blubb %} -a -b +the weak* topology on the dual of $\mathcal{A}$ via the correspondence described above. {% enddefinition %} - +{% proposition %} +The *Gelfand space* of a unital commutative Banach algebra is a compact Hausdorff space. +{% endproposition %} 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 + diff --git a/pages/operator-algebras/operator-topologies.md b/pages/operator-algebras/operator-topologies.md new file mode 100644 index 0000000..2d7722e --- /dev/null +++ b/pages/operator-algebras/operator-topologies.md @@ -0,0 +1,14 @@ +--- +title: Operator Topologies +parent: Operator Algebras +nav_order: 1 +--- + +# {{ page.title }} + +{% definition Weak & Strong Operator Topology %} +TODO +{% enddefinition %} + +{% proof %} +{% endproof %} |