Editing Continuous functional calculus

In [[mathematics]], particularly in [[operator theory]] and [[C*-algebra]] theory, the '''continuous functional calculus''' is a [[functional calculus]] which allows the application of a [[continuous function]] to [[Normal element|normal elements]] of a C*-algebra.

In advanced theory, the applications of this functional calculus are so natural that they are often not even mentioned. It is no overstatement to say that the continuous functional calculus makes ''the'' difference between C*-algebras and general [[Banach algebra|Banach algebras]], in which only a [[holomorphic functional calculus]] exists.

== 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.

== 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}}}}

== 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}}}}

== Applications ==

The following applications are typical and very simple examples of the numerous applications of the continuous functional calculus:

=== Spectrum ===

Let <math>\mathcal{A}</math> be a C*-algebra and <math>a \in \mathcal{A}_N</math> a normal element. Then the following applies to the spectrum {{nowrap|<math>\sigma(a)</math>:{{sfn|Kadison|Ringrose|1983|p=271}}}}

* <math>a</math> is self-adjoint if and only if {{nowrap|<math>\sigma(a) \subseteq \R</math>.}}
* <math>a</math> is unitary if and only if {{nowrap|<math>\sigma(a) \subseteq \mathbb{T} = \{ \lambda \in \C \mid \left\| \lambda \right\| = 1 \}</math>.}}
* <math>a</math> is a projection if and only if {{nowrap|<math>\sigma(a) \subseteq \{ 0, 1 \}</math>.}}

''Proof.''{{sfn|Kadison|Ringrose|1983|p=272}} The continuous functional calculus <math>\Phi_a</math> for the normal element <math>a \in \mathcal{A}</math> is a *-homomorphism with <math>\Phi_a (\operatorname{Id}) = a</math> and thus <math>a</math> is self-adjoint/unitary/a projection if <math>\operatorname{Id} \in C( \sigma(a))</math> is also self-adjoint/unitary/a projection. Exactly then <math>\operatorname{Id}</math> is self-adjoint if <math>z = \text{Id}(z) = \overline{\text{Id}}(z) = \overline{z}</math> holds for all <math>z \in \sigma(a)</math>, i.e. if <math>\sigma(a)</math> is real. Exactly then <math>\text{Id}</math> is unitary if <math>1 = \text{Id}(z) \overline{\operatorname{Id}}(z) = z \overline{z} = |z|^2</math> holds for all <math>z \in \sigma(a)</math>, therefore {{nowrap|<math>\sigma(a) \subseteq \{ \lambda \in \C \ | \ \left\| \lambda \right\| = 1 \}</math>.}} Exactly then <math>\text{Id}</math> is a projection if and only if <math>(\operatorname{Id}(z))^2 = \operatorname{Id}}(z) = \overline{\operatorname{Id}(z)</math>, that is <math>z^2 = z = \overline{z}</math> for all <math>z \in \sigma(a)</math>, i.e. <math>\sigma(a) \subseteq \{ 0,1 \}</math>

=== 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}}}}

=== Absolute value ===

If <math>a \in \mathcal{A}</math>, then the element <math>a^*a</math> is positive, so that the absolute value can be defined by the continuous functional calculus <math>|a| = \sqrt{a^*a}</math>, since it is continuous on the positive real {{nowrap|numbers.{{sfn|Blackadar|2006|pages=64-65}}}}

Let <math>a</math> be a self-adjoint element of a C*-algebra <math>\mathcal{A}</math>, then there exist positive elements <math>a_+,a_- \in \mathcal{A}_+</math>, such that <math>a = a_+ - a_-</math> with <math>a_+ a_- = a_- a_+ = 0</math> holds. The elements <math>a_+</math> and <math>a_-</math> are also referred to as the {{nowrap|[[positive and negative parts]].{{sfn|Kadison|Ringrose|1983|p=246}}}} In addition, <math>|a| = a_+ + a_-</math> {{nowrap|holds.{{sfn|Dixmier|1977|p=15}}}}

''Proof.'' The functions <math>f_+(z) = \max(z,0)</math> and <math>f_-(z) = -\min(z, 0)</math> are continuous functions on <math>\sigma(a) \subseteq \R</math> with <math>\operatorname{Id} (z) = z = f_+(z) -f_-(z)</math> and {{nowrap|<math>f_+(z)f_-(z) = f_-(z)f_+(z) = 0</math>.}} Put <math>a_+ = f_+(a)</math> and <math>a_- = f_-(a)</math>. According to the spectral mapping theorem, <math>a_+</math> and <math>a_-</math> are positive elements for which <math>a = \operatorname{Id}(a) = (f_+ - f_-) (a) = f_+(a) - f_-(a) = a_+ - a_-</math> and <math>a_+ a_- = f_+(a)f_-(a) = (f_+f_-)(a) = 0 = (f_-f_+)(a) = f_-(a)f_+(a) = a_- a_+</math> {{nowrap|holds.{{sfn|Kadison|Ringrose|1983|p=246}}}} Furthermore, <math display="inline">f_+(z) + f_-(z) = |z| = \sqrt{z^* z} = \sqrt{z^2}</math>, such that {{nowrap|<math display="inline">a_+ + a_- = f_+(a) + f_-(a) = |a| = \sqrt{a^* a} = \sqrt{a^2}</math> holds.{{sfn|Dixmier|1977|p=15}}}}

=== 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>.}}

=== Spectral decomposition theorem ===

Let <math>\mathcal{A}</math> be an unital C*-algebra and <math>a \in \mathcal{A}_N</math> a normal element. Let the spectrum consist of <math>n</math> pairwise [[Disjoint sets|disjoint]] [[Closed set|closed]] subsets <math>\sigma_k \subset \C</math> for all <math>1 \leq k \leq n</math>, i.e. {{nowrap|<math>\sigma(a)=\sigma_1 \sqcup \cdots \sqcup \sigma_n</math>.}} Then there exist projections <math>p_1, \ldots, p_n \in \mathcal{A}</math> that have the following properties for all {{nowrap|<math>1 \leq j,k \leq n</math>:{{sfn|Kaballo|2014|p=375}}}}

* For the spectrum, <math>\sigma(p_k) = \sigma_k</math> holds.
* The projections commutate with <math>a</math>, i.e. {{nowrap|<math>p_ka=ap_k</math>.}}
* The projections are [[Orthogonality|orthogonal]], i.e. {{nowrap|<math>p_jp_k=\delta_{jk} p_k</math>.}}
* The sum of the projections is the unit element, i.e. {{nowrap|<math display="inline">\sum_{k=1}^n p_k = e</math>.}}

In particular, there is a decomposition <math display="inline">a = \sum_{k=1}^n a_k</math> for which <math>\sigma(a_k) = \sigma_k</math> holds for all {{nowrap|<math>1 \leq k \leq n</math>.}}

''Proof.''{{sfn|Kaballo|2014|p=375}} Since all <math>\sigma_k</math> are closed, the [[Indicator function|characteristic functions]] <math>\chi_{\sigma_k}</math> are continuous on {{nowrap|<math>\sigma(a)</math>.}} Now let <math>p_k := \chi_{\sigma_k} (a)</math> be defined using the continuous functional. As the <math>\sigma_k</math> are pairwise disjoint, <math>\chi_{\sigma_j} \chi_{\sigma_k} = \delta_{jk} \chi_{\sigma_k}</math> and <math display="inline">\sum_{k=1}^n \chi_{\sigma_k} = \chi_{\cup_{k=1}^n \sigma_k} = \chi_{\sigma(a)} = \textbf{1}</math> holds and thus the <math>p_k</math> satisfy the claimed properties, as can be seen from the properties of the continuous functional equation. For the last statement, let {{nowrap|<math>a_k = a p_k = \operatorname{Id} (a) \cdot \chi_{\sigma_k} (a) = (\operatorname{Id} \cdot \chi_{\sigma_k}) (a)</math>.}}

== Notes ==
{{reflist}}

== References ==
* {{cite book |last=Blackadar |first=Bruce |title=Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. |publisher=Springer |location=Berlin/Heidelberg |year=2006 |isbn=3-540-28486-9}}
* {{cite book |last1=Deitmar |first1=Anton |last2=Echterhoff |first2=Siegfried |title=Principles of Harmonic Analysis. Second Edition. |publisher=Springer |year=2014 |isbn=978-3-319-05791-0}}
* {{cite book |last=Dixmier |first=Jacques |title=Les C*-algèbres et leurs représentations |language=fr |publisher=Gauthier-Villars |year=1969 }}
* {{cite book |last=Dixmier |first=Jacques |title=C*-algebras |publisher=North-Holland |location=Amsterdam/New York/Oxford |year=1977 |isbn=0-7204-0762-1 |translator-last=Jellett |translator-first=Francis }} English translation of {{cite book |display-authors=0 |last=Dixmier |first=Jacques |title=Les C*-algèbres et leurs représentations |language=fr |publisher=Gauthier-Villars |year=1969 }}
* {{cite book |last=Kaballo |first=Winfried |title=Aufbaukurs Funktionalanalysis und Operatortheorie. |language=de |publisher=Springer |location=Berlin/Heidelberg |year=2014 |isbn=978-3-642-37794-5}}
* {{cite book |last1=Kadison |first1=Richard V. |last2=Ringrose |first2=John R. |title=Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. |publisher=Academic Press |location=New York/London |year=1983 |isbn=0-12-393301-3}}
* {{cite book |last=Kaniuth |first=Eberhard |title=A Course in Commutative Banach Algebras. |publisher=Springer |year=2009 |isbn=978-0-387-72475-1}}
* {{cite book |last=Schmüdgen |first=Konrad |title=Unbounded Self-adjoint Operators on Hilbert Space. |publisher=Springer |year=2012 |isbn=978-94-007-4752-4}}
* {{cite book |last1=Reed |first1=Michael |last2=Simon |first2=Barry |title=Methods of modern mathematical physics. vol. 1. Functional analysis |publisher=Academic Press |location=San Diego, CA |year=1980 |isbn=0-12-585050-6}}
* {{cite book |last=Takesaki |first=Masamichi |title=Theory of Operator Algebras I. |publisher=Springer |location=Heidelberg/Berlin |year=1979 |isbn=3-540-90391-7 }}

== External links ==
* [http://planetmath.org/continuousfunctionalcalculus Continuous functional calculus on PlanetMath]

{{SpectralTheory}}

[[Category:Theorems in functional analysis]]
[[Category:C*-algebras]]
[[Category:Functional calculus]]