Editing Continuous functional calculus (section)

=== 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>.}}