Editing Continuous functional calculus (section)

== Theorem ==

{{math theorem | name = continuous functional calculus | math_statement = Let <math>a</math> be a normal element of the C*-algebra <math>\mathcal{A}</math> with [[Identity element|unit element]] <math>e</math> and let <math>C (\sigma(a))</math> be the commutative C*-algebra of continuous functions on <math>\sigma(a)</math>, the spectrum of {{nowrap|<math>a</math>.}} Then there exists exactly one [[*-algebra#*-algebra|*-homomorphism]] <math>\Phi_a \colon C (\sigma(a)) \rightarrow \mathcal{A}</math> with <math>\Phi_a (\boldsymbol{1}) = e</math> for <math>\boldsymbol{1}(z) = 1</math> and <math>\Phi_a(\operatorname{Id}_{\sigma(a)}) = a</math> for the {{nowrap|[[Identity (mathematics)|identity]].{{sfn|Dixmier|1977|pages=12-13}}}}

The mapping <math>\Phi_a</math> is called the continuous functional calculus of the normal element {{nowrap|<math>a</math>.}}
Usually it is suggestively set {{nowrap|<math>f(a) := \Phi_a(f)</math>.{{sfn|Kadison|Ringrose|1983|p=272}}}}}}

Due to the *-homomorphism property, the following calculation rules apply to all functions <math>f,g \in C(\sigma(a))</math> and [[Scalar (mathematics)|scalars]] <math>\lambda,\mu \in \C</math>:{{sfn|Dixmier|1977|p=5,13}}
{|
|
* <math>(\lambda f + \mu g)(a) = \lambda f(a) + \mu g(a) \qquad</math>
|(linear)
|-
|
* <math>(f \cdot g)(a) = f (a) \cdot g(a)</math>
|(multiplicative)
|-
|
* <math>\overline{f}(a) =\colon \; (f^*)(a) = (f(a))^*</math>
|(involutive)
|}
One can therefore imagine actually inserting the normal elements into continuous functions; the obvious algebraic operations behave as expected.

The requirement for a unit element is not a significant restriction. If necessary, a [[Rng (algebra)#Adjoining an identity element (Dorroh extension)|unit element can be adjoined]], yielding the enlarged C*-algebra {{nowrap|<math>\mathcal{A}_1</math>.}} Then if <math>a \in \mathcal{A}</math> and <math>f \in C(\sigma (a))</math> with <math>f(0) = 0</math>, it follows that <math>0 \in \sigma (a)</math> and {{nowrap|<math>f(a)\in \mathcal{A} \subset \mathcal{A}_1</math>.{{sfn|Dixmier|1977|p=14}}}}

The existence and uniqueness of the continuous functional calculus are proven separately:

* ''Existence:'' Since the spectrum of <math>a</math> in the C*-[[subalgebra]] <math>C^*(a,e)</math> generated by <math>a</math> and <math>e</math> is the same as it is in <math>\mathcal{A}</math>, it suffices to show the statement for {{nowrap|<math>\mathcal{A} = C^*(a,e)</math>.{{sfn|Dixmier|1977|p=11}}}} The actual construction is almost immediate from the [[Gelfand representation]]: it suffices to assume <math>\mathcal{A}</math> is the C*-algebra of continuous functions on some compact space <math>X</math> and define {{nowrap|<math>\Phi_a(f) = f \circ x</math>.{{sfn|Dixmier|1977|p=13}}}}

* ''Uniqueness:'' Since <math>\Phi_a(\boldsymbol{1})</math> and <math>\Phi_a(\operatorname{Id}_{\sigma(a)})</math> are fixed, <math>\Phi_a</math> is already uniquely defined for all polynomials <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>, since <math>\Phi_a</math> is a *-homomorphism. These form a [[Dense set|dense]] subalgebra of <math>C(\sigma(a))</math> by the Stone-Weierstrass theorem. Thus <math>\Phi_a</math> is {{nowrap|unique.{{sfn|Dixmier|1977|p=13}}}}

In [[functional analysis]], the continuous functional calculus for a normal operator <math>T</math> is often of interest, i.e. the case where <math>\mathcal{A}</math> is the C*-algebra <math>\mathcal{B}(H)</math> of [[Bounded operator|bounded operators]] on a [[Hilbert space]] {{nowrap|<math>H</math>.}} In the literature, the continuous functional calculus is often only proved for [[Self-adjoint operator|self-adjoint operators]] in this setting. In this case, the proof does not need the Gelfand {{nowrap|representation.{{sfn|Reed|Simon|1980|pages=222-223}}}}