Editing Continuous functional calculus (section)

=== 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}}}}