Editing Continuous functional calculus (section)

=== 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>.}}