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 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 algebras, in which only a holomorphic functional calculus exists.

## Motivation

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If one wants to extend the natural functional calculus for polynomials on the spectrum 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 approximate a continuous function by polynomials according to the Stone-Weierstrass theorem, to insert the element into these polynomials and to show that this sequence of elements converges 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 complex conjugation, which is an involution on the complex numbers.[1] To be able to insert  a {\displaystyle a} in place of  z {\displaystyle z} in this kind of polynomial, Banach *-algebras 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 homomorphism 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 commutative. If  ( p n ( z , z ¯ ) ) n {\displaystyle (p_{n}(z,{\overline {z}}))_{n}} is a sequence of polynomials that converges uniformly 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.

## Theorem

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continuous functional calculus—Let  a {\displaystyle a} be a normal element of the C*-algebra  A {\displaystyle {\mathcal {A}}} with unit element 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 *-homomorphism 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 identity.[2]

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)} .[3]

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 scalars λ , μ ∈ C {\displaystyle \lambda ,\mu \in \mathbb {C} } :[4]

* ( λ f + μ g ) ( a ) = λ f ( a ) + μ g ( a ) {\displaystyle (\lambda f+\mu g)(a)=\lambda f(a)+\mu g(a)\qquad }

| (linear)

|

* ( f ⋅ g ) ( a ) = f ( a ) ⋅ g ( a ) {\displaystyle (f\cdot g)(a)=f(a)\cdot g(a)}

| (multiplicative)

* f ¯ ( a ) = : ( f ∗ ) ( a ) = ( f ( a ) ) ∗ {\displaystyle {\overline {f}}(a)=\colon \;(f^{*})(a)=(f(a))^{*}}

| (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 unit element can be adjoined, 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}} .[5]

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

* Existence: Since the spectrum of  a {\displaystyle a} in the C*-subalgebra 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)} .[6] The actual construction is almost immediate from the Gelfand representation: 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} .[7]

* 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 dense subalgebra of  C ( σ ( a ) ) {\displaystyle C(\sigma (a))} by the Stone-Weierstrass theorem. Thus  Φ a {\displaystyle \Phi _{a}} is unique.[7]

In functional analysis, 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 bounded operators on a Hilbert space H {\displaystyle H} . In the literature, the continuous functional calculus is often only proved for self-adjoint operators in this setting. In this case, the proof does not need the Gelfand representation.[8]

## Further properties of the continuous functional calculus

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The continuous functional calculus  Φ a {\displaystyle \Phi _{a}} is an isometric isomorphism into the C*-subalgebra  C ∗ ( a , e ) {\displaystyle C^{*}(a,e)} generated by  a {\displaystyle a} and  e {\displaystyle e} , that is:[7]

* ‖ 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.
* Φ a ( C ( σ ( a ) ) ) = C ∗ ( a , e ) ⊆ A {\displaystyle \Phi _{a}\left(C(\sigma (a))\right)=C^{*}(a,e)\subseteq {\mathcal {A}}}

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.[9]

The holomorphic functional calculus is extended by the continuous functional calculus in an unambiguous way.[10] 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} } .[3]

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)} .[11] For a power series f ( z ) = ∑ n = 0 ∞ c n z n {\textstyle f(z)=\sum _{n=0}^{\infty }c_{n}z^{n}} , which converges absolutely uniformly 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.[12]

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 composition.[5] 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 inverse function 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} .[13]

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))} .[7]

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)} .[14]

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.[4]

With the spectral mapping theorem, functions with certain properties can be directly related to certain properties of elements of C*-algebras:[15]

* f ( a ) {\displaystyle f(a)} is invertible if and only if  f {\displaystyle f} has no zero on  σ ( a ) {\displaystyle \sigma (a)} .[16] Then  f ( a ) − 1 = 1 f ( a ) {\textstyle f(a)^{-1}={\tfrac {1}{f}}(a)} holds.[17]

* f ( a ) {\displaystyle f(a)} is self-adjoint if and only if  f {\displaystyle f} is real-valued, i.e.  f ( σ ( a ) ) ⊆ R {\displaystyle f(\sigma (a))\subseteq \mathbb {R} } .
* f ( a ) {\displaystyle f(a)} is positive ( 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 )} .
* f ( a ) {\displaystyle f(a)} is unitary if all values of  f {\displaystyle f} lie in the circle group, 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\\}} .
* f ( a ) {\displaystyle f(a)} is a projection 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\\}} .

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} , eigenvectors 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)} .[18]

## Applications

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

### Spectrum

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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)} :[15]

* a {\displaystyle a} is self-adjoint if and only if  σ ( a ) ⊆ R {\displaystyle \sigma (a)\subseteq \mathbb {R} } .
* 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\\}} .
* a {\displaystyle a} is a projection if and only if  σ ( a ) ⊆ { 0 , 1 } {\displaystyle \sigma (a)\subseteq \\{0,1\\}} .

Proof.[3] 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\\}}

### Roots

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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.[19]

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.[19] 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}} .[13]

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} .[20]

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  a α {\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 } .[19]

### Absolute value

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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.[21]

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 positive and negative parts.[22] In addition,  | a | = a + + a − {\displaystyle |a|=a_{+}+a_{-}} holds.[23]

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.[22] 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.[23]

### Unitary elements

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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 imaginary unit. Conversely, if  u ∈ A U {\displaystyle u\in {\mathcal {A}}_{U}} is an unitary element, with the restriction that the spectrum is a proper subset 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}} .[24]

Proof.[24] 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} .

### Spectral decomposition theorem

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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 disjoint closed 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} :[25]

* For the spectrum,  σ ( p k ) = σ k {\displaystyle \sigma (p_{k})=\sigma _{k}} holds.
* The projections commutate with  a {\displaystyle a} , i.e.  p k a = a p k {\displaystyle p_{k}a=ap_{k}} .
* The projections are orthogonal, i.e.  p j p k = δ j k p k {\displaystyle p_{j}p_{k}=\delta _{jk}p_{k}} .
* 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} .

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.[25] Since all  σ k {\displaystyle \sigma _{k}} are closed, the characteristic functions 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)} .

## Notes

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1. ^ Dixmier 1977, p. 3.
2. ^ Dixmier 1977, pp. 12–13.
3. ^ a b c Kadison & Ringrose 1983, p. 272.
4. ^ a b Dixmier 1977, p. 5,13.
5. ^ a b Dixmier 1977, p. 14.
6. ^ Dixmier 1977, p. 11.
7. ^ a b c d Dixmier 1977, p. 13.
8. ^ Reed & Simon 1980, pp. 222–223.
9. ^ Dixmier 1977, pp. 5, 13.
10. ^ Kaniuth 2009, p. 147.
11. ^ Blackadar 2006, p. 62.
12. ^ Deitmar & Echterhoff 2014, p. 55.
13. ^ a b Kadison & Ringrose 1983, p. 275.
14. ^ Kadison & Ringrose 1983, p. 239.
15. ^ a b Kadison & Ringrose 1983, p. 271.
16. ^ Kaballo 2014, p. 332.
17. ^ Schmüdgen 2012, p. 93.
18. ^ Reed & Simon 1980, p. 222.
19. ^ a b c Kadison & Ringrose 1983, pp. 248–249.
20. ^ Blackadar 2006, p. 63.
21. ^ Blackadar 2006, pp. 64–65.
22. ^ a b Kadison & Ringrose 1983, p. 246.
23. ^ a b Dixmier 1977, p. 15.
24. ^ a b Kadison & Ringrose 1983, pp. 274–275.
25. ^ a b Kaballo 2014, p. 375.

## References

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* Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. ISBN 3-540-28486-9.
* Deitmar, Anton; Echterhoff, Siegfried (2014). Principles of Harmonic Analysis. Second Edition. Springer. ISBN 978-3-319-05791-0.
* Dixmier, Jacques (1969). Les C*-algèbres et leurs représentations (in French). Gauthier-Villars.
* Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
* Kaballo, Winfried (2014). Aufbaukurs Funktionalanalysis und Operatortheorie (in German). Berlin/Heidelberg: Springer. ISBN 978-3-642-37794-5.
* 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.
* Kaniuth, Eberhard (2009). A Course in Commutative Banach Algebras. Springer. ISBN 978-0-387-72475-1.
* Schmüdgen, Konrad (2012). Unbounded Self-adjoint Operators on Hilbert Space. Springer. ISBN 978-94-007-4752-4.
* Reed, Michael; Simon, Barry (1980). Methods of modern mathematical physics. vol. 1. Functional analysis. San Diego, CA: Academic Press. ISBN 0-12-585050-6.
* Takesaki, Masamichi (1979). Theory of Operator Algebras I. Heidelberg/Berlin: Springer. ISBN 3-540-90391-7.

## External links

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* Continuous functional calculus on PlanetMath

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Spectral theory and *-algebras

Basic concepts|

* Involution/*-algebra
* Banach algebra
* B*-algebra
* C*-algebra
* Noncommutative topology
* Projection-valued measure
* Spectrum
* Spectrum of a C*-algebra
* Spectral radius
* Operator space

Main results|

* Gelfand–Mazur theorem
* Gelfand–Naimark theorem
* Gelfand representation
* Polar decomposition
* Singular value decomposition
* Spectral theorem
* Spectral theory of normal C*-algebras

Special Elements/Operators|

* Isospectral
* Normal operator
* Hermitian/Self-adjoint operator
* Unitary operator
* Unit

Spectrum|

* Krein–Rutman theorem
* Normal eigenvalue
* Spectrum of a C*-algebra
* Spectral radius
* Spectral asymmetry
* Spectral gap

Decomposition|

* Decomposition of a spectrum
* Continuous
* Point
* Residual
* Approximate point
* Compression
* Direct integral
* Discrete
* Spectral abscissa

Spectral Theorem|

* Borel functional calculus
* Min-max theorem
* Positive operator-valued measure
* Projection-valued measure
* Riesz projector
* Rigged Hilbert space
* Spectral theorem
* Spectral theory of compact operators
* Spectral theory of normal C*-algebras

Special algebras|

* Amenable Banach algebra
* With an Approximate identity
* Banach function algebra
* Disk algebra
* Nuclear C*-algebra
* Uniform algebra
* Von Neumann algebra
* Tomita–Takesaki theory

Finite-Dimensional|

* Alon–Boppana bound
* Bauer–Fike theorem
* Numerical range
* Schur–Horn theorem

Generalizations|

* Dirac spectrum
* Essential spectrum
* Pseudospectrum
* Structure space (Shilov boundary)

Miscellaneous|

* Abstract index group
* Banach algebra cohomology
* Cohen–Hewitt factorization theorem
* Extensions of symmetric operators
* Fredholm theory
* Limiting absorption principle
* Schröder–Bernstein theorems for operator algebras
* Sherman–Takeda theorem
* Unbounded operator

Examples|

* Wiener algebra

Applications|

* Almost Mathieu operator
* Corona theorem
* Hearing the shape of a drum (Dirichlet eigenvalue)
* Heat kernel
* Kuznetsov trace formula
* Lax pair
* Proto-value function
* Ramanujan graph
* Rayleigh–Faber–Krahn inequality
* Spectral geometry
* Spectral method
* Spectral theory of ordinary differential equations
* Sturm–Liouville theory
* Superstrong approximation
* Transfer operator
* Transform theory
* Weyl law
* Wiener–Khinchin theorem

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