Editing Continuous functional calculus (section)

== Further properties of the continuous functional calculus ==

The continuous functional calculus <math>\Phi_a</math> is an [[Isometry|isometric]] [[isomorphism]] into the C*-subalgebra <math>C^*(a,e)</math> generated by <math>a</math> and <math>e</math>, that is:{{sfn|Dixmier|1977|p=13}}

* <math>\left\| \Phi_a(f) \right\| = \left\| f \right\|_{\sigma(a)}</math> for all <math>f \in C(\sigma(a))</math>; <math>\Phi_a</math> is therefore continuous.
* <math>\Phi_a \left( C(\sigma(a)) \right) = C^*(a, e) \subseteq \mathcal{A}</math>

Since <math>a</math> is a normal element of <math>\mathcal{A}</math>, the C*-subalgebra generated by <math>a</math> and <math>e</math> is commutative. In particular, <math>f(a)</math> is normal and all elements of a functional calculus {{nowrap|commutate.{{sfn|Dixmier|1977|pages=5,13}}}}

The [[holomorphic functional calculus]] is [[Restriction (mathematics)#Extension of a function|extended]] by the continuous functional calculus in an unambiguous {{nowrap|way.{{sfn|Kaniuth|2009|p=147}}}} Therefore, for polynomials <math>p(z,\overline{z})</math> the continuous functional calculus corresponds to the natural functional calculus for polynomials: <math display="inline">\Phi_a(p(z, \overline{z})) = p(a, a^*) = \sum_{k,l=0}^N c_{k, l} a^k(a^*)^l</math> for all {{nowrap|<math display="inline">p(z, \overline{z}) = \sum_{k,l=0}^N c_{k,l} z^k\overline{z}^l</math> with <math>c_{k,l} \in \C</math>.{{sfn|Kadison|Ringrose|1983|p=272}}}}

For a sequence of functions <math>f_n \in C(\sigma(a))</math> that converges uniformly on <math>\sigma(a)</math> to a function <math>f \in C(\sigma(a))</math>, <math>f_n(a)</math> converges to {{nowrap|<math>f(a)</math>.}}{{sfn|Blackadar|2006|p=62}} For a [[power series]] <math display="inline">f(z) = \sum_{n=0}^\infty c_n z^n</math>, which converges [[Absolute convergence|absolutely]] [[Uniform convergence|uniformly]] on <math>\sigma(a)</math>, therefore <math display="inline">f(a) = \sum_{n=0}^\infty c_na^n</math> {{nowrap|holds.{{sfn|Deitmar|Echterhoff|2014|p=55}}}}

If <math>f \in \mathcal{C}(\sigma(a))</math> and <math>g\in \mathcal{ C}(\sigma(f(a)))</math>, then <math>(g \circ f)(a) = g(f(a))</math> holds for their {{nowrap|[[Function composition|composition]].{{sfn|Dixmier|1977|p=14}}}} If <math>a,b \in \mathcal{A}_N</math> are two normal elements with <math>f(a) = f(b)</math> and <math>g</math> is the [[inverse function]] of <math>f</math> on both <math>\sigma(a)</math> and <math>\sigma(b)</math>, then <math>a = b</math>, since {{nowrap|<math>a = (f \circ g) (a) = f(g(a)) = f(g(b)) = (f \circ g) (b) = b</math>.{{sfn|Kadison|Ringrose|1983|p=275}}}}

The ''spectral mapping theorem'' applies: <math>\sigma(f(a)) = f(\sigma(a))</math> for all {{nowrap|<math>f \in C(\sigma(a))</math>.{{sfn|Dixmier|1977|p=13}}}}

If <math>ab = ba</math> holds for <math>b \in \mathcal{A}</math>, then <math>f(a)b = bf(a)</math> also holds for all <math>f \in C ( \sigma (a))</math>, i.e. if <math>b</math> commutates with <math>a</math>, then also with the corresponding elements of the continuous functional calculus {{nowrap|<math>f(a)</math>.{{sfn|Kadison|Ringrose|1983|p=239}}}}

Let <math>\Psi \colon \mathcal{A} \rightarrow \mathcal{B}</math> be an unital *-homomorphism between C*-algebras <math>\mathcal{A}</math> and {{nowrap|<math>\mathcal{B}</math>.}} Then <math>\Psi</math> commutates with the continuous functional calculus. The following holds: <math>\Psi(f(a)) = f(\Psi(a))</math> for all {{nowrap|<math>f \in C(\sigma(a))</math>.}} In particular, the continuous functional calculus commutates with the Gelfand {{nowrap|representation.{{sfn|Dixmier|1977|p=5,13}}}}

With the spectral mapping theorem, functions with certain properties can be directly related to certain properties of elements of C*-algebras:{{sfn|Kadison|Ringrose|1983|p=271}}

* <math>f(a)</math> is [[Inverse element|invertible]] if and only if <math>f</math> [[Zero of a function|has no zero]] on {{nowrap|<math>\sigma(a)</math>.{{sfn|Kaballo|2014|p=332}}}} Then <math display="inline">f(a)^{-1} = \tfrac{1}{f} (a)</math> {{nowrap|holds.{{sfn|Schmüdgen|2012|p=93}}}}

* <math>f(a)</math> is [[self-adjoint]] if and only if <math>f</math> is [[Real-valued function|real-valued]], i.e. {{nowrap|<math>f(\sigma(a)) \subseteq \R</math>.}}
* <math>f(a)</math> is [[Positive element|positive]] (<math>f(a) \geq 0</math>) if and only if <math>f \geq 0</math>, i.e. {{nowrap|<math>f(\sigma(a)) \subseteq [0,\infty )</math>.}}
* <math>f(a)</math> is [[Unitary element|unitary]] if all values of <math>f</math> lie in the [[circle group]], i.e. {{nowrap|<math>f(\sigma(a)) \subseteq \mathbb{T} = \{ \lambda \in \C \mid \left\| \lambda \right\| = 1 \}</math>.}}
* <math>f(a)</math> is a [[Projection (mathematics)|projection]] if <math>f</math> only takes on the values <math>0</math> and <math>1</math>, i.e. {{nowrap|<math>f(\sigma(a)) \subseteq \{ 0, 1 \}</math>.}}

These are based on statements about the spectrum of certain elements, which are shown in the Applications section.

In the special case that <math>\mathcal{A}</math> is the C*-algebra of bounded operators <math>\mathcal{B}(H)</math> for a Hilbert space <math>H</math>, [[Eigenvalues and eigenvectors|eigenvectors]] <math>v \in H</math> for the eigenvalue <math>\lambda \in \sigma(T)</math> of a normal operator <math>T \in \mathcal{B}(H)</math> are also eigenvectors for the eigenvalue <math>f(\lambda) \in \sigma(f(T))</math> of the operator {{nowrap|<math>f(T)</math>.}} If <math>Tv = \lambda v</math>, then <math>f(T)v = f(\lambda)v</math> also holds for all {{nowrap|<math>f \in \sigma(T)</math>.{{sfn|Reed|Simon|1980|p=222}}}}