Editing Continuous functional calculus (section)

=== 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}}}}