Continuous functional calculus

<h2 id="motivation">Motivation</h2>
If one wants to extend the <a href="/facts/Functional_calculus/Y84ugVzS">natural functional calculus for polynomials</a> on the <a href="/facts/Banach_algebra/3BTHdpg2">spectrum</a> 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
 of an element 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of a Banach algebra 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 to a functional calculus for continuous functions 
 
 
 
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle C(\sigma (a))}
 
 on the spectrum, it seems obvious to <a href="/facts/Approximation/LvYHsXzq">approximate</a> a continuous function by <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> according to the <a href="/facts/Stone%E2%80%93Weierstrass_theorem/QB07Nwba">Stone-Weierstrass theorem</a>, to insert the element into these polynomials and to show that this <a href="/facts/Sequence/gV2S4uqp">sequence</a> of elements <a href="/facts/Limit_(mathematics)/j8JBhj4e">converges</a> to 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
.
The continuous functions on 
 
 
 
 σ
 (
 a
 )
 ⊂
 
 C
 
 
 
 {\displaystyle \sigma (a)\subset \mathbb {C} }
 
 are approximated by polynomials in 
 
 
 
 z
 
 
 {\displaystyle z}
 
 and 
 
 
 
 
 
 z
 ¯
 
 
 
 
 {\displaystyle {\overline {z}}}
 
, i.e. by polynomials of the form 
 
 
 
 p
 (
 z
 ,
 
 
 z
 ¯
 
 
 )
 =
 
 ∑
 
 k
 ,
 l
 =
 0
 
 
 N
 
 
 
 c
 
 k
 ,
 l
 
 
 
 z
 
 k
 
 
 
 
 
 z
 ¯
 
 
 
 l
 
 
 
 
 (
 
 
 c
 
 k
 ,
 l
 
 
 ∈
 
 C
 
 
 )
 
 
 
 {\textstyle p(z,{\overline {z}})=\sum _{k,l=0}^{N}c_{k,l}z^{k}{\overline {z}}^{l}\;\left(c_{k,l}\in \mathbb {C} \right)}
 
. Here, 
 
 
 
 
 
 z
 ¯
 
 
 
 
 {\displaystyle {\overline {z}}}
 
 denotes the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugation</a>, which is an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a> on the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
To be able to insert 
 
 
 
 a
 
 
 {\displaystyle a}
 
 in place of 
 
 
 
 z
 
 
 {\displaystyle z}
 
 in this kind of polynomial, <a href="/facts/Banach_algebra/3BTHdpg2">Banach *-algebras</a> are considered, i.e. Banach algebras that also have an involution *, and 
 
 
 
 
 a
 
 ∗
 
 
 
 
 {\displaystyle a^{*}}
 
 is inserted in place of 
 
 
 
 
 
 z
 ¯
 
 
 
 
 {\displaystyle {\overline {z}}}
 
. In order to obtain a <a href="/facts/Algebra_homomorphism/b7YL8aPR">homomorphism</a> 
 
 
 
 
 
 C
 
 
 [
 z
 ,
 
 
 z
 ¯
 
 
 ]
 →
 
 
 A
 
 
 
 
 {\displaystyle {\mathbb {C} }[z,{\overline {z}}]\rightarrow {\mathcal {A}}}
 
, a restriction to normal elements, i.e. elements with 
 
 
 
 
 a
 
 ∗
 
 
 a
 =
 a
 
 a
 
 ∗
 
 
 
 
 {\displaystyle a^{*}a=aa^{*}}
 
, is necessary, as the polynomial ring 
 
 
 
 
 C
 
 [
 z
 ,
 
 
 z
 ¯
 
 
 ]
 
 
 {\displaystyle \mathbb {C} [z,{\overline {z}}]}
 
 is <a href="/facts/Commutative_property/WaYxJLxd">commutative</a>.
If 
 
 
 
 (
 
 p
 
 n
 
 
 (
 z
 ,
 
 
 z
 ¯
 
 
 )
 
 )
 
 n
 
 
 
 
 {\displaystyle (p_{n}(z,{\overline {z}}))_{n}}
 
 is a sequence of polynomials that converges <a href="/facts/Uniform_convergence/ecz9eNWn">uniformly</a> on 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
 to a continuous function 
 
 
 
 f
 
 
 {\displaystyle f}
 
, the convergence of the sequence 
 
 
 
 (
 
 p
 
 n
 
 
 (
 a
 ,
 
 a
 
 ∗
 
 
 )
 
 )
 
 n
 
 
 
 
 {\displaystyle (p_{n}(a,a^{*}))_{n}}
 
 in 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 to an element 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 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.

<h2 id="theorem">Theorem</h2>

continuous functional calculus—Let 
 
 
 
 a
 
 
 {\displaystyle a}
 
 be a normal element of the C*-algebra 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 with <a href="/facts/Identity_element/9nss3xHY">unit element</a> 
 
 
 
 e
 
 
 {\displaystyle e}
 
 and let 
 
 
 
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle C(\sigma (a))}
 
 be the commutative C*-algebra of continuous functions on 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
, the spectrum of 
 
 
 
 a
 
 
 {\displaystyle a}
 
. Then there exists exactly one <a href="/facts/*-algebra/vcE5TWnT">*-homomorphism</a> 
 
 
 
 
 Φ
 
 a
 
 
 :
 C
 (
 σ
 (
 a
 )
 )
 →
 
 
 A
 
 
 
 
 {\displaystyle \Phi _{a}\colon C(\sigma (a))\rightarrow {\mathcal {A}}}
 
 with 
 
 
 
 
 Φ
 
 a
 
 
 (
 
 1
 
 )
 =
 e
 
 
 {\displaystyle \Phi _{a}({\boldsymbol {1}})=e}
 
 for 
 
 
 
 
 1
 
 (
 z
 )
 =
 1
 
 
 {\displaystyle {\boldsymbol {1}}(z)=1}
 
 and 
 
 
 
 
 Φ
 
 a
 
 
 (
 
 Id
 
 σ
 (
 a
 )
 
 
 )
 =
 a
 
 
 {\displaystyle \Phi _{a}(\operatorname {Id} _{\sigma (a)})=a}
 
 for the <a href="/facts/Identity_(mathematics)/LR46j4uR">identity</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
The mapping 
 
 
 
 
 Φ
 
 a
 
 
 
 
 {\displaystyle \Phi _{a}}
 
 is called the continuous functional calculus of the normal element 
 
 
 
 a
 
 
 {\displaystyle a}
 
.
Usually it is suggestively set 
 
 
 
 f
 (
 a
 )
 :=
 
 Φ
 
 a
 
 
 (
 f
 )
 
 
 {\displaystyle f(a):=\Phi _{a}(f)}
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

Due to the *-homomorphism property, the following calculation rules apply to all functions 
 
 
 
 f
 ,
 g
 ∈
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle f,g\in C(\sigma (a))}
 
 and <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a> 
 
 
 
 λ
 ,
 μ
 ∈
 
 C
 
 
 
 {\displaystyle \lambda ,\mu \in \mathbb {C} }
 
:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<table><tbody><tr><td><ul><li> ( λ f + μ g ) ( a ) = λ f ( a ) + μ g ( a ) {\displaystyle (\lambda f+\mu g)(a)=\lambda f(a)+\mu g(a)\qquad } </li></ul></td><td>(linear)</td></tr><tr><td><ul><li> ( f ⋅ g ) ( a ) = f ( a ) ⋅ g ( a ) {\displaystyle (f\cdot g)(a)=f(a)\cdot g(a)} </li></ul></td><td>(multiplicative)</td></tr><tr><td><ul><li> f ¯ ( a ) = : ( f ∗ ) ( a ) = ( f ( a ) ) ∗ {\displaystyle {\overline {f}}(a)=\colon \;(f^{*})(a)=(f(a))^{*}} </li></ul></td><td>(involutive)</td></tr></tbody></table>
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 <a href="/facts/Rng_(algebra)/SquNCXKe">unit element can be adjoined</a>, yielding the enlarged C*-algebra 
 
 
 
 
 
 
 A
 
 
 
 1
 
 
 
 
 {\displaystyle {\mathcal {A}}_{1}}
 
. Then if 
 
 
 
 a
 ∈
 
 
 A
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}}
 
 and 
 
 
 
 f
 ∈
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle f\in C(\sigma (a))}
 
 with 
 
 
 
 f
 (
 0
 )
 =
 0
 
 
 {\displaystyle f(0)=0}
 
, it follows that 
 
 
 
 0
 ∈
 σ
 (
 a
 )
 
 
 {\displaystyle 0\in \sigma (a)}
 
 and 
 
 
 
 f
 (
 a
 )
 ∈
 
 
 A
 
 
 ⊂
 
 
 
 A
 
 
 
 1
 
 
 
 
 {\displaystyle f(a)\in {\mathcal {A}}\subset {\mathcal {A}}_{1}}
 
.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
The existence and uniqueness of the continuous functional calculus are proven separately:

<ul><li>Existence: Since the spectrum of 
 
 
 
 a
 
 
 {\displaystyle a}
 
 in the C*-<a href="/facts/Subalgebra/9PEHNF5r">subalgebra</a> 
 
 
 
 
 C
 
 ∗
 
 
 (
 a
 ,
 e
 )
 
 
 {\displaystyle C^{*}(a,e)}
 
 generated by 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 e
 
 
 {\displaystyle e}
 
 is the same as it is in 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
, it suffices to show the statement for 
 
 
 
 
 
 A
 
 
 =
 
 C
 
 ∗
 
 
 (
 a
 ,
 e
 )
 
 
 {\displaystyle {\mathcal {A}}=C^{*}(a,e)}
 
.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> The actual construction is almost immediate from the <a href="/facts/Gelfand_representation/QV54f7Jm">Gelfand representation</a>: it suffices to assume 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 is the C*-algebra of continuous functions on some compact space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and define 
 
 
 
 
 Φ
 
 a
 
 
 (
 f
 )
 =
 f
 ∘
 x
 
 
 {\displaystyle \Phi _{a}(f)=f\circ x}
 
.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li></ul>
<ul><li>Uniqueness: Since 
 
 
 
 
 Φ
 
 a
 
 
 (
 
 1
 
 )
 
 
 {\displaystyle \Phi _{a}({\boldsymbol {1}})}
 
 and 
 
 
 
 
 Φ
 
 a
 
 
 (
 
 Id
 
 σ
 (
 a
 )
 
 
 )
 
 
 {\displaystyle \Phi _{a}(\operatorname {Id} _{\sigma (a)})}
 
 are fixed, 
 
 
 
 
 Φ
 
 a
 
 
 
 
 {\displaystyle \Phi _{a}}
 
 is already uniquely defined for all polynomials 
 
 
 
 p
 (
 z
 ,
 
 
 z
 ¯
 
 
 )
 =
 
 ∑
 
 k
 ,
 l
 =
 0
 
 
 N
 
 
 
 c
 
 k
 ,
 l
 
 
 
 z
 
 k
 
 
 
 
 
 z
 ¯
 
 
 
 l
 
 
 
 
 (
 
 
 c
 
 k
 ,
 l
 
 
 ∈
 
 C
 
 
 )
 
 
 
 {\textstyle p(z,{\overline {z}})=\sum _{k,l=0}^{N}c_{k,l}z^{k}{\overline {z}}^{l}\;\left(c_{k,l}\in \mathbb {C} \right)}
 
, since 
 
 
 
 
 Φ
 
 a
 
 
 
 
 {\displaystyle \Phi _{a}}
 
 is a *-homomorphism. These form a <a href="/facts/Dense_set/nPT9Yxao">dense</a> subalgebra of 
 
 
 
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle C(\sigma (a))}
 
 by the Stone-Weierstrass theorem. Thus 
 
 
 
 
 Φ
 
 a
 
 
 
 
 {\displaystyle \Phi _{a}}
 
 is unique.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li></ul>
In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, the continuous functional calculus for a normal operator 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is often of interest, i.e. the case where 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 is the C*-algebra 
 
 
 
 
 
 B
 
 
 (
 H
 )
 
 
 {\displaystyle {\mathcal {B}}(H)}
 
 of <a href="/facts/Bounded_operator/LYmDTIqR">bounded operators</a> on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> 
 
 
 
 H
 
 
 {\displaystyle H}
 
. In the literature, the continuous functional calculus is often only proved for <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operators</a> in this setting. In this case, the proof does not need the Gelfand representation.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="further-properties-of-the-continuous-functional-calculus">Further properties of the continuous functional calculus</h2>
The continuous functional calculus 
 
 
 
 
 Φ
 
 a
 
 
 
 
 {\displaystyle \Phi _{a}}
 
 is an <a href="/facts/Isometry/K5wX8ah6">isometric</a> <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> into the C*-subalgebra 
 
 
 
 
 C
 
 ∗
 
 
 (
 a
 ,
 e
 )
 
 
 {\displaystyle C^{*}(a,e)}
 
 generated by 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 e
 
 
 {\displaystyle e}
 
, that is:<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<ul><li>
 
 
 
 
 ‖
 
 
 Φ
 
 a
 
 
 (
 f
 )
 
 ‖
 
 =
 
 
 ‖
 f
 ‖
 
 
 σ
 (
 a
 )
 
 
 
 
 {\displaystyle \left\|\Phi _{a}(f)\right\|=\left\|f\right\|_{\sigma (a)}}
 
 for all 
 
 
 
 f
 ∈
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle f\in C(\sigma (a))}
 
; 
 
 
 
 
 Φ
 
 a
 
 
 
 
 {\displaystyle \Phi _{a}}
 
 is therefore continuous.</li>
<li>
 
 
 
 
 Φ
 
 a
 
 
 
 (
 
 C
 (
 σ
 (
 a
 )
 )
 
 )
 
 =
 
 C
 
 ∗
 
 
 (
 a
 ,
 e
 )
 ⊆
 
 
 A
 
 
 
 
 {\displaystyle \Phi _{a}\left(C(\sigma (a))\right)=C^{*}(a,e)\subseteq {\mathcal {A}}}
 
</li></ul>
Since 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is a normal element of 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
, the C*-subalgebra generated by 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 e
 
 
 {\displaystyle e}
 
 is commutative. In particular, 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 is normal and all elements of a functional calculus commutate.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
The <a href="/facts/Holomorphic_functional_calculus/xtWI9f4V">holomorphic functional calculus</a> is <a href="/facts/Restriction_(mathematics)/JV59DhKy">extended</a> by the continuous functional calculus in an unambiguous way.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> Therefore, for polynomials 
 
 
 
 p
 (
 z
 ,
 
 
 z
 ¯
 
 
 )
 
 
 {\displaystyle p(z,{\overline {z}})}
 
 the continuous functional calculus corresponds to the natural functional calculus for polynomials: 
 
 
 
 
 Φ
 
 a
 
 
 (
 p
 (
 z
 ,
 
 
 z
 ¯
 
 
 )
 )
 =
 p
 (
 a
 ,
 
 a
 
 ∗
 
 
 )
 =
 
 ∑
 
 k
 ,
 l
 =
 0
 
 
 N
 
 
 
 c
 
 k
 ,
 l
 
 
 
 a
 
 k
 
 
 (
 
 a
 
 ∗
 
 
 
 )
 
 l
 
 
 
 
 {\textstyle \Phi _{a}(p(z,{\overline {z}}))=p(a,a^{*})=\sum _{k,l=0}^{N}c_{k,l}a^{k}(a^{*})^{l}}
 
 for all 
 
 
 
 p
 (
 z
 ,
 
 
 z
 ¯
 
 
 )
 =
 
 ∑
 
 k
 ,
 l
 =
 0
 
 
 N
 
 
 
 c
 
 k
 ,
 l
 
 
 
 z
 
 k
 
 
 
 
 
 z
 ¯
 
 
 
 l
 
 
 
 
 {\textstyle p(z,{\overline {z}})=\sum _{k,l=0}^{N}c_{k,l}z^{k}{\overline {z}}^{l}}
 
 with 
 
 
 
 
 c
 
 k
 ,
 l
 
 
 ∈
 
 C
 
 
 
 {\displaystyle c_{k,l}\in \mathbb {C} }
 
.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
For a sequence of functions 
 
 
 
 
 f
 
 n
 
 
 ∈
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle f_{n}\in C(\sigma (a))}
 
 that converges uniformly on 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
 to a function 
 
 
 
 f
 ∈
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle f\in C(\sigma (a))}
 
, 
 
 
 
 
 f
 
 n
 
 
 (
 a
 )
 
 
 {\displaystyle f_{n}(a)}
 
 converges to 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> For a <a href="/facts/Power_series/vYh6sTps">power series</a> 
 
 
 
 f
 (
 z
 )
 =
 
 ∑
 
 n
 =
 0
 
 
 ∞
 
 
 
 c
 
 n
 
 
 
 z
 
 n
 
 
 
 
 {\textstyle f(z)=\sum _{n=0}^{\infty }c_{n}z^{n}}
 
, which converges <a href="/facts/Absolute_convergence/BXz9kBJ5">absolutely</a> <a href="/facts/Uniform_convergence/ecz9eNWn">uniformly</a> on 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
, therefore 
 
 
 
 f
 (
 a
 )
 =
 
 ∑
 
 n
 =
 0
 
 
 ∞
 
 
 
 c
 
 n
 
 
 
 a
 
 n
 
 
 
 
 {\textstyle f(a)=\sum _{n=0}^{\infty }c_{n}a^{n}}
 
 holds.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
If 
 
 
 
 f
 ∈
 
 
 C
 
 
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle f\in {\mathcal {C}}(\sigma (a))}
 
 and 
 
 
 
 g
 ∈
 
 
 C
 
 
 (
 σ
 (
 f
 (
 a
 )
 )
 )
 
 
 {\displaystyle g\in {\mathcal {C}}(\sigma (f(a)))}
 
, then 
 
 
 
 (
 g
 ∘
 f
 )
 (
 a
 )
 =
 g
 (
 f
 (
 a
 )
 )
 
 
 {\displaystyle (g\circ f)(a)=g(f(a))}
 
 holds for their <a href="/facts/Function_composition/r5Pvnmth">composition</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> If 
 
 
 
 a
 ,
 b
 ∈
 
 
 
 A
 
 
 
 N
 
 
 
 
 {\displaystyle a,b\in {\mathcal {A}}_{N}}
 
 are two normal elements with 
 
 
 
 f
 (
 a
 )
 =
 f
 (
 b
 )
 
 
 {\displaystyle f(a)=f(b)}
 
 and 
 
 
 
 g
 
 
 {\displaystyle g}
 
 is the <a href="/facts/Inverse_function/Tg5nnXlu">inverse function</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 on both 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
 and 
 
 
 
 σ
 (
 b
 )
 
 
 {\displaystyle \sigma (b)}
 
, then 
 
 
 
 a
 =
 b
 
 
 {\displaystyle a=b}
 
, since 
 
 
 
 a
 =
 (
 f
 ∘
 g
 )
 (
 a
 )
 =
 f
 (
 g
 (
 a
 )
 )
 =
 f
 (
 g
 (
 b
 )
 )
 =
 (
 f
 ∘
 g
 )
 (
 b
 )
 =
 b
 
 
 {\displaystyle a=(f\circ g)(a)=f(g(a))=f(g(b))=(f\circ g)(b)=b}
 
.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
The spectral mapping theorem applies: 
 
 
 
 σ
 (
 f
 (
 a
 )
 )
 =
 f
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle \sigma (f(a))=f(\sigma (a))}
 
 for all 
 
 
 
 f
 ∈
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle f\in C(\sigma (a))}
 
.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
If 
 
 
 
 a
 b
 =
 b
 a
 
 
 {\displaystyle ab=ba}
 
 holds for 
 
 
 
 b
 ∈
 
 
 A
 
 
 
 
 {\displaystyle b\in {\mathcal {A}}}
 
, then 
 
 
 
 f
 (
 a
 )
 b
 =
 b
 f
 (
 a
 )
 
 
 {\displaystyle f(a)b=bf(a)}
 
 also holds for all 
 
 
 
 f
 ∈
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle f\in C(\sigma (a))}
 
, i.e. if 
 
 
 
 b
 
 
 {\displaystyle b}
 
 commutates with 
 
 
 
 a
 
 
 {\displaystyle a}
 
, then also with the corresponding elements of the continuous functional calculus 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>
Let 
 
 
 
 Ψ
 :
 
 
 A
 
 
 →
 
 
 B
 
 
 
 
 {\displaystyle \Psi \colon {\mathcal {A}}\rightarrow {\mathcal {B}}}
 
 be an unital *-homomorphism between C*-algebras 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 and 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
. Then 
 
 
 
 Ψ
 
 
 {\displaystyle \Psi }
 
 commutates with the continuous functional calculus. The following holds: 
 
 
 
 Ψ
 (
 f
 (
 a
 )
 )
 =
 f
 (
 Ψ
 (
 a
 )
 )
 
 
 {\displaystyle \Psi (f(a))=f(\Psi (a))}
 
 for all 
 
 
 
 f
 ∈
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle f\in C(\sigma (a))}
 
. In particular, the continuous functional calculus commutates with the Gelfand representation.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>
With the spectral mapping theorem, functions with certain properties can be directly related to certain properties of elements of C*-algebras:<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>

<ul><li>
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 is <a href="/facts/Inverse_element/CwMygHSj">invertible</a> if and only if 
 
 
 
 f
 
 
 {\displaystyle f}
 
 <a href="/facts/Zero_of_a_function/mlK3dhoT">has no zero</a> on 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> Then 
 
 
 
 f
 (
 a
 
 )
 
 −
 1
 
 
 =
 
 
 
 1
 f
 
 
 
 (
 a
 )
 
 
 {\textstyle f(a)^{-1}={\tfrac {1}{f}}(a)}
 
 holds.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a></li></ul>
<ul><li>
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 is <a href="/facts/Self-adjoint/rKqGtlpy">self-adjoint</a> if and only if 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is <a href="/facts/Real-valued_function/jetqlJpK">real-valued</a>, i.e. 
 
 
 
 f
 (
 σ
 (
 a
 )
 )
 ⊆
 
 R
 
 
 
 {\displaystyle f(\sigma (a))\subseteq \mathbb {R} }
 
.</li>
<li>
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 is <a href="/facts/Positive_element/kvTXwFHj">positive</a> (
 
 
 
 f
 (
 a
 )
 ≥
 0
 
 
 {\displaystyle f(a)\geq 0}
 
) if and only if 
 
 
 
 f
 ≥
 0
 
 
 {\displaystyle f\geq 0}
 
, i.e. 
 
 
 
 f
 (
 σ
 (
 a
 )
 )
 ⊆
 [
 0
 ,
 ∞
 )
 
 
 {\displaystyle f(\sigma (a))\subseteq [0,\infty )}
 
.</li>
<li>
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 is <a href="/facts/Unitary_element/N6zYf8XN">unitary</a> if all values of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 lie in the <a href="/facts/Circle_group/kuzdWtHo">circle group</a>, i.e. 
 
 
 
 f
 (
 σ
 (
 a
 )
 )
 ⊆
 
 T
 
 =
 {
 λ
 ∈
 
 C
 
 ∣
 
 ‖
 λ
 ‖
 
 =
 1
 }
 
 
 {\displaystyle f(\sigma (a))\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid \left\|\lambda \right\|=1\}}
 
.</li>
<li>
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 is a <a href="/facts/Projection_(mathematics)/QxxC9XTp">projection</a> if 
 
 
 
 f
 
 
 {\displaystyle f}
 
 only takes on the values 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 and 
 
 
 
 1
 
 
 {\displaystyle 1}
 
, i.e. 
 
 
 
 f
 (
 σ
 (
 a
 )
 )
 ⊆
 {
 0
 ,
 1
 }
 
 
 {\displaystyle f(\sigma (a))\subseteq \{0,1\}}
 
.</li></ul>
These are based on statements about the spectrum of certain elements, which are shown in the Applications section.
In the special case that 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 is the C*-algebra of bounded operators 
 
 
 
 
 
 B
 
 
 (
 H
 )
 
 
 {\displaystyle {\mathcal {B}}(H)}
 
 for a Hilbert space 
 
 
 
 H
 
 
 {\displaystyle H}
 
, <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvectors</a> 
 
 
 
 v
 ∈
 H
 
 
 {\displaystyle v\in H}
 
 for the eigenvalue 
 
 
 
 λ
 ∈
 σ
 (
 T
 )
 
 
 {\displaystyle \lambda \in \sigma (T)}
 
 of a normal operator 
 
 
 
 T
 ∈
 
 
 B
 
 
 (
 H
 )
 
 
 {\displaystyle T\in {\mathcal {B}}(H)}
 
 are also eigenvectors for the eigenvalue 
 
 
 
 f
 (
 λ
 )
 ∈
 σ
 (
 f
 (
 T
 )
 )
 
 
 {\displaystyle f(\lambda )\in \sigma (f(T))}
 
 of the operator 
 
 
 
 f
 (
 T
 )
 
 
 {\displaystyle f(T)}
 
. If 
 
 
 
 T
 v
 =
 λ
 v
 
 
 {\displaystyle Tv=\lambda v}
 
, then 
 
 
 
 f
 (
 T
 )
 v
 =
 f
 (
 λ
 )
 v
 
 
 {\displaystyle f(T)v=f(\lambda )v}
 
 also holds for all 
 
 
 
 f
 ∈
 σ
 (
 T
 )
 
 
 {\displaystyle f\in \sigma (T)}
 
.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>

<h2 id="applications">Applications</h2>
The following applications are typical and very simple examples of the numerous applications of the continuous functional calculus:

<h3>Spectrum</h3>
Let 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 be a C*-algebra and 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 N
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{N}}
 
 a normal element. Then the following applies to the spectrum 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
:<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>

<ul><li>
 
 
 
 a
 
 
 {\displaystyle a}
 
 is self-adjoint if and only if 
 
 
 
 σ
 (
 a
 )
 ⊆
 
 R
 
 
 
 {\displaystyle \sigma (a)\subseteq \mathbb {R} }
 
.</li>
<li>
 
 
 
 a
 
 
 {\displaystyle a}
 
 is unitary if and only if 
 
 
 
 σ
 (
 a
 )
 ⊆
 
 T
 
 =
 {
 λ
 ∈
 
 C
 
 ∣
 
 ‖
 λ
 ‖
 
 =
 1
 }
 
 
 {\displaystyle \sigma (a)\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid \left\|\lambda \right\|=1\}}
 
.</li>
<li>
 
 
 
 a
 
 
 {\displaystyle a}
 
 is a projection if and only if 
 
 
 
 σ
 (
 a
 )
 ⊆
 {
 0
 ,
 1
 }
 
 
 {\displaystyle \sigma (a)\subseteq \{0,1\}}
 
.</li></ul>
Proof.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> The continuous functional calculus 
 
 
 
 
 Φ
 
 a
 
 
 
 
 {\displaystyle \Phi _{a}}
 
 for the normal element 
 
 
 
 a
 ∈
 
 
 A
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}}
 
 is a *-homomorphism with 
 
 
 
 
 Φ
 
 a
 
 
 (
 Id
 )
 =
 a
 
 
 {\displaystyle \Phi _{a}(\operatorname {Id} )=a}
 
 and thus 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is self-adjoint/unitary/a projection if 
 
 
 
 Id
 ∈
 C
 (
 σ
 (
 a
 )
 )
 
 
 {\displaystyle \operatorname {Id} \in C(\sigma (a))}
 
 is also self-adjoint/unitary/a projection. Exactly then 
 
 
 
 Id
 
 
 {\displaystyle \operatorname {Id} }
 
 is self-adjoint if 
 
 
 
 z
 =
 
 Id
 
 (
 z
 )
 =
 
 
 Id
 ¯
 
 
 (
 z
 )
 =
 
 
 z
 ¯
 
 
 
 
 {\displaystyle z={\text{Id}}(z)={\overline {\text{Id}}}(z)={\overline {z}}}
 
 holds for all 
 
 
 
 z
 ∈
 σ
 (
 a
 )
 
 
 {\displaystyle z\in \sigma (a)}
 
, i.e. if 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
 is real. Exactly then 
 
 
 
 
 Id
 
 
 
 {\displaystyle {\text{Id}}}
 
 is unitary if 
 
 
 
 1
 =
 
 Id
 
 (
 z
 )
 
 
 Id
 ¯
 
 
 (
 z
 )
 =
 z
 
 
 z
 ¯
 
 
 =
 
 |
 
 z
 
 
 |
 
 
 2
 
 
 
 
 {\displaystyle 1={\text{Id}}(z){\overline {\operatorname {Id} }}(z)=z{\overline {z}}=|z|^{2}}
 
 holds for all 
 
 
 
 z
 ∈
 σ
 (
 a
 )
 
 
 {\displaystyle z\in \sigma (a)}
 
, therefore 
 
 
 
 σ
 (
 a
 )
 ⊆
 {
 λ
 ∈
 
 C
 
  
 
 |
 
  
 
 ‖
 λ
 ‖
 
 =
 1
 }
 
 
 {\displaystyle \sigma (a)\subseteq \{\lambda \in \mathbb {C} \ |\ \left\|\lambda \right\|=1\}}
 
. Exactly then 
 
 
 
 
 Id
 
 
 
 {\displaystyle {\text{Id}}}
 
 is a projection if and only if 
 
 
 
 
 (
 Id
 ⁡
 (
 z
 )
 
 )
 
 2
 
 
 =
 Id
 
 
 (
 z
 )
 =
 
 
 
 Id
 ⁡
 (
 z
 )
 
 ¯
 
 
 
 {\displaystyle (\operatorname {Id} (z))^{2}=\operatorname {Id} }(z)={\overline {\operatorname {Id} (z)}}
 
, that is 
 
 
 
 
 z
 
 2
 
 
 =
 z
 =
 
 
 z
 ¯
 
 
 
 
 {\displaystyle z^{2}=z={\overline {z}}}
 
 for all 
 
 
 
 z
 ∈
 σ
 (
 a
 )
 
 
 {\displaystyle z\in \sigma (a)}
 
, i.e. 
 
 
 
 σ
 (
 a
 )
 ⊆
 {
 0
 ,
 1
 }
 
 
 {\displaystyle \sigma (a)\subseteq \{0,1\}}

<h3>Roots</h3>
Let 
 
 
 
 a
 
 
 {\displaystyle a}
 
 be a positive element of a C*-algebra 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
. Then for every 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
 there exists a uniquely determined positive element 
 
 
 
 b
 ∈
 
 
 
 A
 
 
 
 +
 
 
 
 
 {\displaystyle b\in {\mathcal {A}}_{+}}
 
 with 
 
 
 
 
 b
 
 n
 
 
 =
 a
 
 
 {\displaystyle b^{n}=a}
 
, i.e. a unique 
 
 
 
 n
 
 
 {\displaystyle n}
 
-th root.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a>
Proof. For each 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
, the root function 
 
 
 
 
 f
 
 n
 
 
 :
 
 
 R
 
 
 0
 
 
 +
 
 
 →
 
 
 R
 
 
 0
 
 
 +
 
 
 ,
 x
 ↦
 
 
 x
 
 n
 
 
 
 
 
 {\displaystyle f_{n}\colon \mathbb {R} _{0}^{+}\to \mathbb {R} _{0}^{+},x\mapsto {\sqrt[{n}]{x}}}
 
 is a continuous function on 
 
 
 
 σ
 (
 a
 )
 ⊆
 
 
 R
 
 
 0
 
 
 +
 
 
 
 
 {\displaystyle \sigma (a)\subseteq \mathbb {R} _{0}^{+}}
 
. If 
 
 
 
 b
 
 :
 =
 
 f
 
 n
 
 
 (
 a
 )
 
 
 {\displaystyle b\;\colon =f_{n}(a)}
 
 is defined using the continuous functional calculus, then 
 
 
 
 
 b
 
 n
 
 
 =
 (
 
 f
 
 n
 
 
 (
 a
 )
 
 )
 
 n
 
 
 =
 (
 
 f
 
 n
 
 
 n
 
 
 )
 (
 a
 )
 =
 
 Id
 
 σ
 (
 a
 )
 
 
 ⁡
 (
 a
 )
 =
 a
 
 
 {\displaystyle b^{n}=(f_{n}(a))^{n}=(f_{n}^{n})(a)=\operatorname {Id} _{\sigma (a)}(a)=a}
 
 follows from the properties of the calculus. From the spectral mapping theorem follows 
 
 
 
 σ
 (
 b
 )
 =
 σ
 (
 
 f
 
 n
 
 
 (
 a
 )
 )
 =
 
 f
 
 n
 
 
 (
 σ
 (
 a
 )
 )
 ⊆
 [
 0
 ,
 ∞
 )
 
 
 {\displaystyle \sigma (b)=\sigma (f_{n}(a))=f_{n}(\sigma (a))\subseteq [0,\infty )}
 
, i.e. 
 
 
 
 b
 
 
 {\displaystyle b}
 
 is positive.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> If 
 
 
 
 c
 ∈
 
 
 
 A
 
 
 
 +
 
 
 
 
 {\displaystyle c\in {\mathcal {A}}_{+}}
 
 is another positive element with 
 
 
 
 
 c
 
 n
 
 
 =
 a
 =
 
 b
 
 n
 
 
 
 
 {\displaystyle c^{n}=a=b^{n}}
 
, then 
 
 
 
 c
 =
 
 f
 
 n
 
 
 (
 
 c
 
 n
 
 
 )
 =
 
 f
 
 n
 
 
 (
 
 b
 
 n
 
 
 )
 =
 b
 
 
 {\displaystyle c=f_{n}(c^{n})=f_{n}(b^{n})=b}
 
 holds, as the root function on the positive real numbers is an inverse function to the function 
 
 
 
 z
 ↦
 
 z
 
 n
 
 
 
 
 {\displaystyle z\mapsto z^{n}}
 
.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a>
If 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{sa}}
 
 is a self-adjoint element, then at least for every odd 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
 there is a uniquely determined self-adjoint element 
 
 
 
 b
 ∈
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle b\in {\mathcal {A}}_{sa}}
 
 with 
 
 
 
 
 b
 
 n
 
 
 =
 a
 
 
 {\displaystyle b^{n}=a}
 
.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>
Similarly, for a positive element 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of a C*-algebra 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
, each 
 
 
 
 α
 ≥
 0
 
 
 {\displaystyle \alpha \geq 0}
 
 defines a uniquely determined positive element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle a^\alpha}
 of 
 
 
 
 
 C
 
 ∗
 
 
 (
 a
 )
 
 
 {\displaystyle C^{*}(a)}
 
, such that 
 
 
 
 
 a
 
 α
 
 
 
 a
 
 β
 
 
 =
 
 a
 
 α
 +
 β
 
 
 
 
 {\displaystyle a^{\alpha }a^{\beta }=a^{\alpha +\beta }}
 
 holds for all 
 
 
 
 α
 ,
 β
 ≥
 0
 
 
 {\displaystyle \alpha ,\beta \geq 0}
 
. If 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is invertible, this can also be extended to negative values of 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a>

<h3>Absolute value</h3>
If 
 
 
 
 a
 ∈
 
 
 A
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}}
 
, then the element 
 
 
 
 
 a
 
 ∗
 
 
 a
 
 
 {\displaystyle a^{*}a}
 
 is positive, so that the absolute value can be defined by the continuous functional calculus 
 
 
 
 
 |
 
 a
 
 |
 
 =
 
 
 
 a
 
 ∗
 
 
 a
 
 
 
 
 {\displaystyle |a|={\sqrt {a^{*}a}}}
 
, since it is continuous on the positive real numbers.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a>
Let 
 
 
 
 a
 
 
 {\displaystyle a}
 
 be a self-adjoint element of a C*-algebra 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
, then there exist positive elements 
 
 
 
 
 a
 
 +
 
 
 ,
 
 a
 
 −
 
 
 ∈
 
 
 
 A
 
 
 
 +
 
 
 
 
 {\displaystyle a_{+},a_{-}\in {\mathcal {A}}_{+}}
 
, such that 
 
 
 
 a
 =
 
 a
 
 +
 
 
 −
 
 a
 
 −
 
 
 
 
 {\displaystyle a=a_{+}-a_{-}}
 
 with 
 
 
 
 
 a
 
 +
 
 
 
 a
 
 −
 
 
 =
 
 a
 
 −
 
 
 
 a
 
 +
 
 
 =
 0
 
 
 {\displaystyle a_{+}a_{-}=a_{-}a_{+}=0}
 
 holds. The elements 
 
 
 
 
 a
 
 +
 
 
 
 
 {\displaystyle a_{+}}
 
 and 
 
 
 
 
 a
 
 −
 
 
 
 
 {\displaystyle a_{-}}
 
 are also referred to as the <a href="/facts/Positive_and_negative_parts/Pzy06lhk">positive and negative parts</a>.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> In addition, 
 
 
 
 
 |
 
 a
 
 |
 
 =
 
 a
 
 +
 
 
 +
 
 a
 
 −
 
 
 
 
 {\displaystyle |a|=a_{+}+a_{-}}
 
 holds.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>
Proof. The functions 
 
 
 
 
 f
 
 +
 
 
 (
 z
 )
 =
 max
 (
 z
 ,
 0
 )
 
 
 {\displaystyle f_{+}(z)=\max(z,0)}
 
 and 
 
 
 
 
 f
 
 −
 
 
 (
 z
 )
 =
 −
 min
 (
 z
 ,
 0
 )
 
 
 {\displaystyle f_{-}(z)=-\min(z,0)}
 
 are continuous functions on 
 
 
 
 σ
 (
 a
 )
 ⊆
 
 R
 
 
 
 {\displaystyle \sigma (a)\subseteq \mathbb {R} }
 
 with 
 
 
 
 Id
 ⁡
 (
 z
 )
 =
 z
 =
 
 f
 
 +
 
 
 (
 z
 )
 −
 
 f
 
 −
 
 
 (
 z
 )
 
 
 {\displaystyle \operatorname {Id} (z)=z=f_{+}(z)-f_{-}(z)}
 
 and 
 
 
 
 
 f
 
 +
 
 
 (
 z
 )
 
 f
 
 −
 
 
 (
 z
 )
 =
 
 f
 
 −
 
 
 (
 z
 )
 
 f
 
 +
 
 
 (
 z
 )
 =
 0
 
 
 {\displaystyle f_{+}(z)f_{-}(z)=f_{-}(z)f_{+}(z)=0}
 
. Put 
 
 
 
 
 a
 
 +
 
 
 =
 
 f
 
 +
 
 
 (
 a
 )
 
 
 {\displaystyle a_{+}=f_{+}(a)}
 
 and 
 
 
 
 
 a
 
 −
 
 
 =
 
 f
 
 −
 
 
 (
 a
 )
 
 
 {\displaystyle a_{-}=f_{-}(a)}
 
. According to the spectral mapping theorem, 
 
 
 
 
 a
 
 +
 
 
 
 
 {\displaystyle a_{+}}
 
 and 
 
 
 
 
 a
 
 −
 
 
 
 
 {\displaystyle a_{-}}
 
 are positive elements for which 
 
 
 
 a
 =
 Id
 ⁡
 (
 a
 )
 =
 (
 
 f
 
 +
 
 
 −
 
 f
 
 −
 
 
 )
 (
 a
 )
 =
 
 f
 
 +
 
 
 (
 a
 )
 −
 
 f
 
 −
 
 
 (
 a
 )
 =
 
 a
 
 +
 
 
 −
 
 a
 
 −
 
 
 
 
 {\displaystyle a=\operatorname {Id} (a)=(f_{+}-f_{-})(a)=f_{+}(a)-f_{-}(a)=a_{+}-a_{-}}
 
 and 
 
 
 
 
 a
 
 +
 
 
 
 a
 
 −
 
 
 =
 
 f
 
 +
 
 
 (
 a
 )
 
 f
 
 −
 
 
 (
 a
 )
 =
 (
 
 f
 
 +
 
 
 
 f
 
 −
 
 
 )
 (
 a
 )
 =
 0
 =
 (
 
 f
 
 −
 
 
 
 f
 
 +
 
 
 )
 (
 a
 )
 =
 
 f
 
 −
 
 
 (
 a
 )
 
 f
 
 +
 
 
 (
 a
 )
 =
 
 a
 
 −
 
 
 
 a
 
 +
 
 
 
 
 {\displaystyle a_{+}a_{-}=f_{+}(a)f_{-}(a)=(f_{+}f_{-})(a)=0=(f_{-}f_{+})(a)=f_{-}(a)f_{+}(a)=a_{-}a_{+}}
 
 holds.<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a> Furthermore, 
 
 
 
 
 f
 
 +
 
 
 (
 z
 )
 +
 
 f
 
 −
 
 
 (
 z
 )
 =
 
 |
 
 z
 
 |
 
 =
 
 
 
 z
 
 ∗
 
 
 z
 
 
 =
 
 
 
 z
 
 2
 
 
 
 
 
 
 {\textstyle f_{+}(z)+f_{-}(z)=|z|={\sqrt {z^{*}z}}={\sqrt {z^{2}}}}
 
, such that 
 
 
 
 
 a
 
 +
 
 
 +
 
 a
 
 −
 
 
 =
 
 f
 
 +
 
 
 (
 a
 )
 +
 
 f
 
 −
 
 
 (
 a
 )
 =
 
 |
 
 a
 
 |
 
 =
 
 
 
 a
 
 ∗
 
 
 a
 
 
 =
 
 
 
 a
 
 2
 
 
 
 
 
 
 {\textstyle a_{+}+a_{-}=f_{+}(a)+f_{-}(a)=|a|={\sqrt {a^{*}a}}={\sqrt {a^{2}}}}
 
 holds.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a>

<h3>Unitary elements</h3>
If 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is a self-adjoint element of a C*-algebra 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 with unit element 
 
 
 
 e
 
 
 {\displaystyle e}
 
, then 
 
 
 
 u
 =
 
 
 e
 
 
 
 i
 
 a
 
 
 
 
 {\displaystyle u=\mathrm {e} ^{\mathrm {i} a}}
 
 is unitary, where 
 
 
 
 
 i
 
 
 
 {\displaystyle \mathrm {i} }
 
 denotes the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>. Conversely, if 
 
 
 
 u
 ∈
 
 
 
 A
 
 
 
 U
 
 
 
 
 {\displaystyle u\in {\mathcal {A}}_{U}}
 
 is an unitary element, with the restriction that the spectrum is a <a href="/facts/Subset/SJGERmbA">proper subset</a> of the unit circle, i.e. 
 
 
 
 σ
 (
 u
 )
 ⊊
 
 T
 
 
 
 {\displaystyle \sigma (u)\subsetneq \mathbb {T} }
 
, there exists a self-adjoint element 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 s
 a
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{sa}}
 
 with 
 
 
 
 u
 =
 
 
 e
 
 
 
 i
 
 a
 
 
 
 
 {\displaystyle u=\mathrm {e} ^{\mathrm {i} a}}
 
.<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a>
Proof.<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a> It is 
 
 
 
 u
 =
 f
 (
 a
 )
 
 
 {\displaystyle u=f(a)}
 
 with 
 
 
 
 f
 :
 
 R
 
 →
 
 C
 
 ,
  
 x
 ↦
 
 
 e
 
 
 
 i
 
 x
 
 
 
 
 {\displaystyle f\colon \mathbb {R} \to \mathbb {C} ,\ x\mapsto \mathrm {e} ^{\mathrm {i} x}}
 
, since 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is self-adjoint, it follows that 
 
 
 
 σ
 (
 a
 )
 ⊂
 
 R
 
 
 
 {\displaystyle \sigma (a)\subset \mathbb {R} }
 
, i.e. 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a function on the spectrum of 
 
 
 
 a
 
 
 {\displaystyle a}
 
. Since 
 
 
 
 f
 ⋅
 
 
 f
 ¯
 
 
 =
 
 
 f
 ¯
 
 
 ⋅
 f
 =
 1
 
 
 {\displaystyle f\cdot {\overline {f}}={\overline {f}}\cdot f=1}
 
, using the functional calculus 
 
 
 
 u
 
 u
 
 ∗
 
 
 =
 
 u
 
 ∗
 
 
 u
 =
 e
 
 
 {\displaystyle uu^{*}=u^{*}u=e}
 
 follows, i.e. 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is unitary. Since for the other statement there is a 
 
 
 
 
 z
 
 0
 
 
 ∈
 
 T
 
 
 
 {\displaystyle z_{0}\in \mathbb {T} }
 
, such that 
 
 
 
 σ
 (
 u
 )
 ⊆
 {
 
 
 e
 
 
 
 i
 
 z
 
 
 ∣
 
 z
 
 0
 
 
 ≤
 z
 ≤
 
 z
 
 0
 
 
 +
 2
 π
 }
 
 
 {\displaystyle \sigma (u)\subseteq \{\mathrm {e} ^{\mathrm {i} z}\mid z_{0}\leq z\leq z_{0}+2\pi \}}
 
 the function 
 
 
 
 f
 (
 
 
 e
 
 
 
 i
 
 z
 
 
 )
 =
 z
 
 
 {\displaystyle f(\mathrm {e} ^{\mathrm {i} z})=z}
 
 is a real-valued continuous function on the spectrum 
 
 
 
 σ
 (
 u
 )
 
 
 {\displaystyle \sigma (u)}
 
 for 
 
 
 
 
 z
 
 0
 
 
 ≤
 z
 ≤
 
 z
 
 0
 
 
 +
 2
 π
 
 
 {\displaystyle z_{0}\leq z\leq z_{0}+2\pi }
 
, such that 
 
 
 
 a
 =
 f
 (
 u
 )
 
 
 {\displaystyle a=f(u)}
 
 is a self-adjoint element that satisfies 
 
 
 
 
 
 e
 
 
 
 i
 
 a
 
 
 =
 
 
 e
 
 
 
 i
 
 f
 (
 u
 )
 
 
 =
 u
 
 
 {\displaystyle \mathrm {e} ^{\mathrm {i} a}=\mathrm {e} ^{\mathrm {i} f(u)}=u}
 
.

<h3>Spectral decomposition theorem</h3>
Let 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 be an unital C*-algebra and 
 
 
 
 a
 ∈
 
 
 
 A
 
 
 
 N
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}_{N}}
 
 a normal element. Let the spectrum consist of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 pairwise <a href="/facts/Disjoint_sets/nLr8XRVd">disjoint</a> <a href="/facts/Closed_set/upYnRTUj">closed</a> subsets 
 
 
 
 
 σ
 
 k
 
 
 ⊂
 
 C
 
 
 
 {\displaystyle \sigma _{k}\subset \mathbb {C} }
 
 for all 
 
 
 
 1
 ≤
 k
 ≤
 n
 
 
 {\displaystyle 1\leq k\leq n}
 
, i.e. 
 
 
 
 σ
 (
 a
 )
 =
 
 σ
 
 1
 
 
 ⊔
 ⋯
 ⊔
 
 σ
 
 n
 
 
 
 
 {\displaystyle \sigma (a)=\sigma _{1}\sqcup \cdots \sqcup \sigma _{n}}
 
. Then there exist projections 
 
 
 
 
 p
 
 1
 
 
 ,
 …
 ,
 
 p
 
 n
 
 
 ∈
 
 
 A
 
 
 
 
 {\displaystyle p_{1},\ldots ,p_{n}\in {\mathcal {A}}}
 
 that have the following properties for all 
 
 
 
 1
 ≤
 j
 ,
 k
 ≤
 n
 
 
 {\displaystyle 1\leq j,k\leq n}
 
:<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a>

<ul><li>For the spectrum, 
 
 
 
 σ
 (
 
 p
 
 k
 
 
 )
 =
 
 σ
 
 k
 
 
 
 
 {\displaystyle \sigma (p_{k})=\sigma _{k}}
 
 holds.</li>
<li>The projections commutate with 
 
 
 
 a
 
 
 {\displaystyle a}
 
, i.e. 
 
 
 
 
 p
 
 k
 
 
 a
 =
 a
 
 p
 
 k
 
 
 
 
 {\displaystyle p_{k}a=ap_{k}}
 
.</li>
<li>The projections are <a href="/facts/Orthogonality/JVIjYPog">orthogonal</a>, i.e. 
 
 
 
 
 p
 
 j
 
 
 
 p
 
 k
 
 
 =
 
 δ
 
 j
 k
 
 
 
 p
 
 k
 
 
 
 
 {\displaystyle p_{j}p_{k}=\delta _{jk}p_{k}}
 
.</li>
<li>The sum of the projections is the unit element, i.e. 
 
 
 
 
 ∑
 
 k
 =
 1
 
 
 n
 
 
 
 p
 
 k
 
 
 =
 e
 
 
 {\textstyle \sum _{k=1}^{n}p_{k}=e}
 
.</li></ul>
In particular, there is a decomposition 
 
 
 
 a
 =
 
 ∑
 
 k
 =
 1
 
 
 n
 
 
 
 a
 
 k
 
 
 
 
 {\textstyle a=\sum _{k=1}^{n}a_{k}}
 
 for which 
 
 
 
 σ
 (
 
 a
 
 k
 
 
 )
 =
 
 σ
 
 k
 
 
 
 
 {\displaystyle \sigma (a_{k})=\sigma _{k}}
 
 holds for all 
 
 
 
 1
 ≤
 k
 ≤
 n
 
 
 {\displaystyle 1\leq k\leq n}
 
.
Proof.<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a> Since all 
 
 
 
 
 σ
 
 k
 
 
 
 
 {\displaystyle \sigma _{k}}
 
 are closed, the <a href="/facts/Indicator_function/QEuh04NM">characteristic functions</a> 
 
 
 
 
 χ
 
 
 σ
 
 k
 
 
 
 
 
 
 {\displaystyle \chi _{\sigma _{k}}}
 
 are continuous on 
 
 
 
 σ
 (
 a
 )
 
 
 {\displaystyle \sigma (a)}
 
. Now let 
 
 
 
 
 p
 
 k
 
 
 :=
 
 χ
 
 
 σ
 
 k
 
 
 
 
 (
 a
 )
 
 
 {\displaystyle p_{k}:=\chi _{\sigma _{k}}(a)}
 
 be defined using the continuous functional. As the 
 
 
 
 
 σ
 
 k
 
 
 
 
 {\displaystyle \sigma _{k}}
 
 are pairwise disjoint, 
 
 
 
 
 χ
 
 
 σ
 
 j
 
 
 
 
 
 χ
 
 
 σ
 
 k
 
 
 
 
 =
 
 δ
 
 j
 k
 
 
 
 χ
 
 
 σ
 
 k
 
 
 
 
 
 
 {\displaystyle \chi _{\sigma _{j}}\chi _{\sigma _{k}}=\delta _{jk}\chi _{\sigma _{k}}}
 
 and 
 
 
 
 
 ∑
 
 k
 =
 1
 
 
 n
 
 
 
 χ
 
 
 σ
 
 k
 
 
 
 
 =
 
 χ
 
 
 ∪
 
 k
 =
 1
 
 
 n
 
 
 
 σ
 
 k
 
 
 
 
 =
 
 χ
 
 σ
 (
 a
 )
 
 
 =
 
 
 1
 
 
 
 
 {\textstyle \sum _{k=1}^{n}\chi _{\sigma _{k}}=\chi _{\cup _{k=1}^{n}\sigma _{k}}=\chi _{\sigma (a)}={\textbf {1}}}
 
 holds and thus the 
 
 
 
 
 p
 
 k
 
 
 
 
 {\displaystyle p_{k}}
 
 satisfy the claimed properties, as can be seen from the properties of the continuous functional equation. For the last statement, let 
 
 
 
 
 a
 
 k
 
 
 =
 a
 
 p
 
 k
 
 
 =
 Id
 ⁡
 (
 a
 )
 ⋅
 
 χ
 
 
 σ
 
 k
 
 
 
 
 (
 a
 )
 =
 (
 Id
 ⋅
 
 χ
 
 
 σ
 
 k
 
 
 
 
 )
 (
 a
 )
 
 
 {\displaystyle a_{k}=ap_{k}=\operatorname {Id} (a)\cdot \chi _{\sigma _{k}}(a)=(\operatorname {Id} \cdot \chi _{\sigma _{k}})(a)}
 
.

<h2 id="notes">Notes</h2>

<ul><li>Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-28486-9.</li>
<li>Deitmar, Anton; Echterhoff, Siegfried (2014). Principles of Harmonic Analysis. Second Edition. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-05791-0.</li>
<li>Dixmier, Jacques (1969). Les C*-algèbres et leurs représentations (in French). Gauthier-Villars.</li>
<li>Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.</li>
<li>Kaballo, Winfried (2014). Aufbaukurs Funktionalanalysis und Operatortheorie (in German). Berlin/Heidelberg: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-37794-5.</li>
<li>Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-393301-3.</li>
<li>Kaniuth, Eberhard (2009). A Course in Commutative Banach Algebras. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-72475-1.</li>
<li>Schmüdgen, Konrad (2012). Unbounded Self-adjoint Operators on Hilbert Space. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-94-007-4752-4.</li>
<li>Reed, Michael; Simon, Barry (1980). Methods of modern mathematical physics. vol. 1. Functional analysis. San Diego, CA: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-585050-6.</li>
<li>Takesaki, Masamichi (1979). Theory of Operator Algebras I. Heidelberg/Berlin: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-90391-7.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://planetmath.org/continuousfunctionalcalculus">Continuous functional calculus on PlanetMath</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Dixmier 1977, p. 3. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Dixmier 1977, pp. 12–13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kadison & Ringrose 1983, p. 272. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Dixmier 1977, p. 5,13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Dixmier 1977, p. 14. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Dixmier 1977, p. 11. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Dixmier 1977, p. 13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Dixmier 1977, p. 13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Reed & Simon 1980, pp. 222–223. - Reed, Michael; Simon, Barry (1980). Methods of modern mathematical physics. vol. 1. Functional analysis. San Diego, CA: Academic Press. ISBN 0-12-585050-6. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Dixmier 1977, p. 13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Dixmier 1977, pp. 5, 13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Kaniuth 2009, p. 147. - Kaniuth, Eberhard (2009). A Course in Commutative Banach Algebras. Springer. ISBN 978-0-387-72475-1. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Kadison & Ringrose 1983, p. 272. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Blackadar 2006, p. 62. - Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. ISBN 3-540-28486-9. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Deitmar & Echterhoff 2014, p. 55. - Deitmar, Anton; Echterhoff, Siegfried (2014). Principles of Harmonic Analysis. Second Edition. Springer. ISBN 978-3-319-05791-0. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Dixmier 1977, p. 14. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Kadison & Ringrose 1983, p. 275. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Dixmier 1977, p. 13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Kadison & Ringrose 1983, p. 239. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Dixmier 1977, p. 5,13. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Kadison & Ringrose 1983, p. 271. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Kaballo 2014, p. 332. - Kaballo, Winfried (2014). Aufbaukurs Funktionalanalysis und Operatortheorie (in German). Berlin/Heidelberg: Springer. ISBN 978-3-642-37794-5. <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Schmüdgen 2012, p. 93. - Schmüdgen, Konrad (2012). Unbounded Self-adjoint Operators on Hilbert Space. Springer. ISBN 978-94-007-4752-4. <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Reed & Simon 1980, p. 222. - Reed, Michael; Simon, Barry (1980). Methods of modern mathematical physics. vol. 1. Functional analysis. San Diego, CA: Academic Press. ISBN 0-12-585050-6. <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Kadison & Ringrose 1983, p. 271. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Kadison & Ringrose 1983, p. 272. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Kadison & Ringrose 1983, pp. 248–249. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Kadison & Ringrose 1983, pp. 248–249. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Kadison & Ringrose 1983, p. 275. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Blackadar 2006, p. 63. - Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. ISBN 3-540-28486-9. <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Kadison & Ringrose 1983, pp. 248–249. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Blackadar 2006, pp. 64–65. - Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. ISBN 3-540-28486-9. <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Kadison & Ringrose 1983, p. 246. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Dixmier 1977, p. 15. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Kadison & Ringrose 1983, p. 246. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Dixmier 1977, p. 15. - Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Kadison & Ringrose 1983, pp. 274–275. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Kadison & Ringrose 1983, pp. 274–275. - Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3. <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Kaballo 2014, p. 375. - Kaballo, Winfried (2014). Aufbaukurs Funktionalanalysis und Operatortheorie (in German). Berlin/Heidelberg: Springer. ISBN 978-3-642-37794-5. <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Kaballo 2014, p. 375. - Kaballo, Winfried (2014). Aufbaukurs Funktionalanalysis und Operatortheorie (in German). Berlin/Heidelberg: Springer. ISBN 978-3-642-37794-5. <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
</ol>

Continuous functional calculus open-in-new

Continuous functional calculus