Editing Continuous functional calculus (section)

== Applications ==

The following applications are typical and very simple examples of the numerous applications of the continuous functional calculus:

=== Spectrum ===

Let <math>\mathcal{A}</math> be a C*-algebra and <math>a \in \mathcal{A}_N</math> a normal element. Then the following applies to the spectrum {{nowrap|<math>\sigma(a)</math>:{{sfn|Kadison|Ringrose|1983|p=271}}}}

* <math>a</math> is self-adjoint if and only if {{nowrap|<math>\sigma(a) \subseteq \R</math>.}}
* <math>a</math> is unitary if and only if {{nowrap|<math>\sigma(a) \subseteq \mathbb{T} = \{ \lambda \in \C \mid \left\| \lambda \right\| = 1 \}</math>.}}
* <math>a</math> is a projection if and only if {{nowrap|<math>\sigma(a) \subseteq \{ 0, 1 \}</math>.}}

''Proof.''{{sfn|Kadison|Ringrose|1983|p=272}} The continuous functional calculus <math>\Phi_a</math> for the normal element <math>a \in \mathcal{A}</math> is a *-homomorphism with <math>\Phi_a (\operatorname{Id}) = a</math> and thus <math>a</math> is self-adjoint/unitary/a projection if <math>\operatorname{Id} \in C( \sigma(a))</math> is also self-adjoint/unitary/a projection. Exactly then <math>\operatorname{Id}</math> is self-adjoint if <math>z = \text{Id}(z) = \overline{\text{Id}}(z) = \overline{z}</math> holds for all <math>z \in \sigma(a)</math>, i.e. if <math>\sigma(a)</math> is real. Exactly then <math>\text{Id}</math> is unitary if <math>1 = \text{Id}(z) \overline{\operatorname{Id}}(z) = z \overline{z} = |z|^2</math> holds for all <math>z \in \sigma(a)</math>, therefore {{nowrap|<math>\sigma(a) \subseteq \{ \lambda \in \C \ | \ \left\| \lambda \right\| = 1 \}</math>.}} Exactly then <math>\text{Id}</math> is a projection if and only if <math>(\operatorname{Id}(z))^2 = \operatorname{Id}}(z) = \overline{\operatorname{Id}(z)</math>, that is <math>z^2 = z = \overline{z}</math> for all <math>z \in \sigma(a)</math>, i.e. <math>\sigma(a) \subseteq \{ 0,1 \}</math>

=== Roots ===

Let <math>a</math> be a positive element of a C*-algebra {{nowrap|<math>\mathcal{A}</math>.}} Then for every <math>n \in \mathbb{N}</math> there exists a uniquely determined positive element <math>b \in \mathcal{A}_+</math> with <math>b^n =a</math>, i.e. a unique <math>n</math>-th {{nowrap|root.{{sfn|Kadison|Ringrose|1983|pages=248-249}}}}

''Proof.'' For each <math>n \in \mathbb{N}</math>, the root function <math>f_n \colon \R_0^+ \to \R_0^+, x \mapsto \sqrt[n]x</math> is a continuous function on {{nowrap|<math>\sigma (a) \subseteq \R_0^+</math>.}} If <math>b \; \colon = f_n (a)</math> is defined using the continuous functional calculus, then <math>b^n = (f_n(a))^n = (f_n^n)(a) = \operatorname{Id}_{\sigma(a)}(a)=a</math> follows from the properties of the calculus. From the spectral mapping theorem follows <math>\sigma(b) = \sigma(f_n(a)) = f_n(\sigma(a)) \subseteq [0,\infty)</math>, i.e. <math>b</math> is {{nowrap|positive.{{sfn|Kadison|Ringrose|1983|pages=248-249}}}} If <math>c \in \mathcal{A}_+</math> is another positive element with <math>c^n = a = b^n</math>, then <math>c = f_n (c^n) = f_n(b^n) = b</math> holds, as the root function on the positive real numbers is an inverse function to the function {{nowrap|<math>z \mapsto z^n</math>.{{sfn|Kadison|Ringrose|1983|p=275}}}}

If <math>a \in \mathcal{A}_{sa}</math> is a self-adjoint element, then at least for every odd <math>n \in \N</math> there is a uniquely determined self-adjoint element <math>b \in \mathcal{A}_{sa}</math> with {{nowrap|<math>b^n = a</math>.{{sfn|Blackadar|2006|p=63}}}}

Similarly, for a positive element <math>a</math> of a C*-algebra <math>\mathcal{A}</math>, each <math>\alpha \geq 0</math> defines a uniquely determined positive element <math>a^\alpha</math> of <math>C^*(a)</math>, such that <math>a^\alpha a^\beta = a^{\alpha + \beta}</math> holds for all {{nowrap|<math>\alpha, \beta \geq 0</math>.}} If <math>a</math> is invertible, this can also be extended to negative values of {{nowrap|<math>\alpha</math>.{{sfn|Kadison|Ringrose|1983|pages=248-249}}}}

=== Absolute value ===

If <math>a \in \mathcal{A}</math>, then the element <math>a^*a</math> is positive, so that the absolute value can be defined by the continuous functional calculus <math>|a| = \sqrt{a^*a}</math>, since it is continuous on the positive real {{nowrap|numbers.{{sfn|Blackadar|2006|pages=64-65}}}}

Let <math>a</math> be a self-adjoint element of a C*-algebra <math>\mathcal{A}</math>, then there exist positive elements <math>a_+,a_- \in \mathcal{A}_+</math>, such that <math>a = a_+ - a_-</math> with <math>a_+ a_- = a_- a_+ = 0</math> holds. The elements <math>a_+</math> and <math>a_-</math> are also referred to as the {{nowrap|[[positive and negative parts]].{{sfn|Kadison|Ringrose|1983|p=246}}}} In addition, <math>|a| = a_+ + a_-</math> {{nowrap|holds.{{sfn|Dixmier|1977|p=15}}}}

''Proof.'' The functions <math>f_+(z) = \max(z,0)</math> and <math>f_-(z) = -\min(z, 0)</math> are continuous functions on <math>\sigma(a) \subseteq \R</math> with <math>\operatorname{Id} (z) = z = f_+(z) -f_-(z)</math> and {{nowrap|<math>f_+(z)f_-(z) = f_-(z)f_+(z) = 0</math>.}} Put <math>a_+ = f_+(a)</math> and <math>a_- = f_-(a)</math>. According to the spectral mapping theorem, <math>a_+</math> and <math>a_-</math> are positive elements for which <math>a = \operatorname{Id}(a) = (f_+ - f_-) (a) = f_+(a) - f_-(a) = a_+ - a_-</math> and <math>a_+ a_- = f_+(a)f_-(a) = (f_+f_-)(a) = 0 = (f_-f_+)(a) = f_-(a)f_+(a) = a_- a_+</math> {{nowrap|holds.{{sfn|Kadison|Ringrose|1983|p=246}}}} Furthermore, <math display="inline">f_+(z) + f_-(z) = |z| = \sqrt{z^* z} = \sqrt{z^2}</math>, such that {{nowrap|<math display="inline">a_+ + a_- = f_+(a) + f_-(a) = |a| = \sqrt{a^* a} = \sqrt{a^2}</math> holds.{{sfn|Dixmier|1977|p=15}}}}

=== Unitary elements ===

If <math>a</math> is a self-adjoint element of a C*-algebra <math>\mathcal{A}</math> with unit element <math>e</math>, then <math>u = \mathrm{e}^{\mathrm{i} a}</math> is unitary, where <math>\mathrm{i}</math> denotes the [[imaginary unit]]. Conversely, if <math>u \in \mathcal{A}_U</math> is an unitary element, with the restriction that the spectrum is a [[Subset|proper subset]] of the unit circle, i.e. <math>\sigma(u) \subsetneq \mathbb{T}</math>, there exists a self-adjoint element <math>a \in \mathcal{A}_{sa}</math> with {{nowrap|<math>u = \mathrm{e}^{\mathrm{i} a}</math>.{{sfn|Kadison|Ringrose|1983|pages=274-275}}}}

''Proof.''{{sfn|Kadison|Ringrose|1983|pages=274-275}} It is <math>u = f(a)</math> with <math>f \colon \R \to \C,\ x \mapsto \mathrm{e}^{\mathrm{i}x}</math>, since <math>a</math> is self-adjoint, it follows that <math>\sigma(a) \subset \R</math>, i.e. <math>f</math> is a function on the spectrum of {{nowrap|<math>a</math>.}} Since <math>f\cdot \overline{f} = \overline{f}\cdot f = 1</math>, using the functional calculus <math>uu^* = u^*u = e</math> follows, i.e. <math>u</math> is unitary. Since for the other statement there is a <math>z_0 \in \mathbb{T}</math>, such that <math>\sigma(u) \subseteq \{ \mathrm{e}^{\mathrm{i} z} \mid z_0 \leq z \leq z_0 + 2 \pi \}</math> the function <math>f(\mathrm{e}^{\mathrm{i} z}) = z</math> is a real-valued continuous function on the spectrum <math>\sigma(u)</math> for <math>z_0 \leq z \leq z_0 + 2 \pi</math>, such that <math>a = f(u)</math> is a self-adjoint element that satisfies {{nowrap|<math>\mathrm{e}^{\mathrm{i} a} = \mathrm{e}^{\mathrm{i} f(u)} = u</math>.}}

=== Spectral decomposition theorem ===

Let <math>\mathcal{A}</math> be an unital C*-algebra and <math>a \in \mathcal{A}_N</math> a normal element. Let the spectrum consist of <math>n</math> pairwise [[Disjoint sets|disjoint]] [[Closed set|closed]] subsets <math>\sigma_k \subset \C</math> for all <math>1 \leq k \leq n</math>, i.e. {{nowrap|<math>\sigma(a)=\sigma_1 \sqcup \cdots \sqcup \sigma_n</math>.}} Then there exist projections <math>p_1, \ldots, p_n \in \mathcal{A}</math> that have the following properties for all {{nowrap|<math>1 \leq j,k \leq n</math>:{{sfn|Kaballo|2014|p=375}}}}

* For the spectrum, <math>\sigma(p_k) = \sigma_k</math> holds.
* The projections commutate with <math>a</math>, i.e. {{nowrap|<math>p_ka=ap_k</math>.}}
* The projections are [[Orthogonality|orthogonal]], i.e. {{nowrap|<math>p_jp_k=\delta_{jk} p_k</math>.}}
* The sum of the projections is the unit element, i.e. {{nowrap|<math display="inline">\sum_{k=1}^n p_k = e</math>.}}

In particular, there is a decomposition <math display="inline">a = \sum_{k=1}^n a_k</math> for which <math>\sigma(a_k) = \sigma_k</math> holds for all {{nowrap|<math>1 \leq k \leq n</math>.}}

''Proof.''{{sfn|Kaballo|2014|p=375}} Since all <math>\sigma_k</math> are closed, the [[Indicator function|characteristic functions]] <math>\chi_{\sigma_k}</math> are continuous on {{nowrap|<math>\sigma(a)</math>.}} Now let <math>p_k := \chi_{\sigma_k} (a)</math> be defined using the continuous functional. As the <math>\sigma_k</math> are pairwise disjoint, <math>\chi_{\sigma_j} \chi_{\sigma_k} = \delta_{jk} \chi_{\sigma_k}</math> and <math display="inline">\sum_{k=1}^n \chi_{\sigma_k} = \chi_{\cup_{k=1}^n \sigma_k} = \chi_{\sigma(a)} = \textbf{1}</math> holds and thus the <math>p_k</math> satisfy the claimed properties, as can be seen from the properties of the continuous functional equation. For the last statement, let {{nowrap|<math>a_k = a p_k = \operatorname{Id} (a) \cdot \chi_{\sigma_k} (a) = (\operatorname{Id} \cdot \chi_{\sigma_k}) (a)</math>.}}