Editing Continuous functional calculus (section)

== Motivation ==

If one wants to extend the [[Functional calculus|natural functional calculus for polynomials]] on the [[Banach algebra#Spectral theory|spectrum]] <math>\sigma(a)</math> of an element <math>a</math> of a Banach algebra <math>\mathcal{A}</math> to a functional calculus for continuous functions <math>C(\sigma(a))</math> on the spectrum, it seems obvious to [[Approximation|approximate]] a continuous function by [[Polynomial|polynomials]] according to the [[Stone–Weierstrass theorem|Stone-Weierstrass theorem]], to insert the element into these polynomials and to show that this [[sequence]] of elements [[Limit (mathematics)|converges]] to {{nowrap|<math>\mathcal{A}</math>.}}
The continuous functions on <math>\sigma(a) \subset \C</math> are approximated by polynomials in <math>z</math> and <math>\overline{z}</math>, i.e. by polynomials of the form {{nowrap|<math display="inline">p(z, \overline{z}) = \sum_{k,l=0}^N c_{k,l} z^k\overline{z}^l \; \left( c_{k,l} \in \C \right)</math>.}} Here, <math>\overline{z}</math> denotes the [[Complex conjugate|complex conjugation]], which is an [[Involution (mathematics)|involution]] on the {{nowrap|[[Complex number|complex numbers]].{{sfn|Dixmier|1977|p=3}}}}
To be able to insert <math>a</math> in place of <math>z</math> in this kind of polynomial, [[Banach algebra#Banach *-algebras|Banach *-algebras]] are considered, i.e. Banach algebras that also have an involution *, and <math>a^*</math> is inserted in place of {{nowrap|<math>\overline{z}</math>.}} In order to obtain a [[Algebra homomorphism|homomorphism]] <math>{\mathbb C}[z,\overline{z}]\rightarrow\mathcal{A}</math>, a restriction to normal elements, i.e. elements with <math>a^*a = aa^*</math>, is necessary, as the polynomial ring <math>\C[z,\overline{z}]</math> is [[Commutative property|commutative]].
If <math>(p_n(z,\overline{z}))_n</math> is a sequence of polynomials that converges [[Uniform convergence|uniformly]] on <math>\sigma(a)</math> to a continuous function <math>f</math>, the convergence of the sequence <math>(p_n(a,a^*))_n</math> in <math>\mathcal{A}</math> to an element <math>f(a)</math> must be ensured. A detailed analysis of this convergence problem shows that it is necessary to resort to C*-algebras. These considerations lead to the so-called continuous functional calculus.