Inmathematics, particularly inoperator theory andC*-algebra theory, thecontinuous functional calculus is afunctional calculus which allows the application of acontinuous function tonormal 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 makesthe difference between C*-algebras and generalBanach algebras, in which only aholomorphic functional calculus exists.

If one wants to extend thenatural functional calculus for polynomials on thespectrum${\displaystyle \sigma (a)}$ of an element${\displaystyle a}$ of a Banach algebra${\displaystyle \mathcal{A}}$ to a functional calculus for continuous functions${\displaystyle C(\sigma (a))}$ on the spectrum, it seems obvious toapproximate a continuous function bypolynomials according to theStone-Weierstrass theorem, to insert the element into these polynomials and to show that thissequence of elementsconverges to${\displaystyle \mathcal{A}}$.The continuous functions on${\displaystyle \sigma (a)\subset \mathbb{C}}$ are approximated by polynomials in${\displaystyle z}$ and${\displaystyle \overline{z}}$, i.e. by polynomials of the form$p(z,\overline{z})=\sum _{k,l=0}^N{c}_{k,l}{z}^k{\overline{z}}^l\left(c_{k,l}\in \mathbb{C}\right)$. Here,${\displaystyle \overline{z}}$ denotes thecomplex conjugation, which is aninvolution on thecomplex numbers.^[1]To be able to insert${\displaystyle a}$ in place of${\displaystyle z}$ in this kind of polynomial,Banach *-algebras are considered, i.e. Banach algebras that also have an involution *, and${\displaystyle a^{\ast }}$ is inserted in place of${\displaystyle \overline{z}}$. In order to obtain ahomomorphism${\displaystyle \mathbb{C}[z,\overline{z}]\to \mathcal{A}}$, a restriction to normal elements, i.e. elements with${\displaystyle a^{\ast }a=a{a}^{\ast }}$, is necessary, as the polynomial ring${\displaystyle \mathbb{C}[z,\overline{z}]}$ iscommutative.If${\displaystyle (p_n(z,\overline{z}){)}_n}$ is a sequence of polynomials that convergesuniformly on${\displaystyle \sigma (a)}$ to a continuous function${\displaystyle f}$, the convergence of the sequence${\displaystyle (p_n(a,a^{\ast }){)}_n}$ in${\displaystyle \mathcal{A}}$ to an element${\displaystyle f(a)}$ 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.

Due to the *-homomorphism property, the following calculation rules apply to all functions${\displaystyle f,g\in C(\sigma (a))}$ andscalars${\displaystyle \lambda ,\mu \in \mathbb{C}}$:^[4]

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, aunit element can be adjoined, yielding the enlarged C*-algebra${\displaystyle {\mathcal{A}}_1}$. Then if${\displaystyle a\in \mathcal{A}}$ and${\displaystyle f\in C(\sigma (a))}$ with${\displaystyle f(0)=0}$, it follows that${\displaystyle 0\in \sigma (a)}$ and${\displaystyle f(a)\in \mathcal{A}\subset {\mathcal{A}}_1}$.^[5]

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

• Existence: Since the spectrum of${\displaystyle a}$ in the C*-subalgebra${\displaystyle C^{\ast }(a,e)}$ generated by${\displaystyle a}$ and${\displaystyle e}$ is the same as it is in${\displaystyle \mathcal{A}}$, it suffices to show the statement for${\displaystyle \mathcal{A}=C^{\ast }(a,e)}$.^[6] The actual construction is almost immediate from theGelfand representation: it suffices to assume${\displaystyle \mathcal{A}}$ is the C*-algebra of continuous functions on some compact space${\displaystyle X}$ and define${\displaystyle {\mathrm{\Phi }}_a(f)=f\circ x}$.^[7]

• Uniqueness: Since${\displaystyle {\mathrm{\Phi }}_a(\mathbf{1})}$ and${\displaystyle {\mathrm{\Phi }}_a({\mathrm{Id}}_{\sigma (a)})}$ are fixed,${\displaystyle {\mathrm{\Phi }}_a}$ is already uniquely defined for all polynomials$p(z,\overline{z})=\sum _{k,l=0}^N{c}_{k,l}{z}^k{\overline{z}}^l\left(c_{k,l}\in \mathbb{C}\right)$, since${\displaystyle {\mathrm{\Phi }}_a}$ is a *-homomorphism. These form adense subalgebra of${\displaystyle C(\sigma (a))}$ by the Stone-Weierstrass theorem. Thus${\displaystyle {\mathrm{\Phi }}_a}$ isunique.^[7]

Infunctional analysis, the continuous functional calculus for a normal operator${\displaystyle T}$ is often of interest, i.e. the case where${\displaystyle \mathcal{A}}$ is the C*-algebra${\displaystyle \mathcal{B}(H)}$ ofbounded operators on aHilbert space${\displaystyle H}$. In the literature, the continuous functional calculus is often only proved forself-adjoint operators in this setting. In this case, the proof does not need the Gelfandrepresentation.^[8]

The continuous functional calculus${\displaystyle {\mathrm{\Phi }}_a}$ is anisometricisomorphism into the C*-subalgebra${\displaystyle C^{\ast }(a,e)}$ generated by${\displaystyle a}$ and${\displaystyle e}$, that is:^[7]

• ${\displaystyle \Vert {\mathrm{\Phi }}_a(f)\Vert ={\Vert f\Vert }_{\sigma (a)}}$ for all${\displaystyle f\in C(\sigma (a))}$;${\displaystyle {\mathrm{\Phi }}_a}$ is therefore continuous.
• ${\displaystyle {\mathrm{\Phi }}_a\left(C(\sigma (a))\right)=C^{\ast }(a,e)\subseteq \mathcal{A}}$

Since${\displaystyle a}$ is a normal element of${\displaystyle \mathcal{A}}$, the C*-subalgebra generated by${\displaystyle a}$ and${\displaystyle e}$ is commutative. In particular,${\displaystyle f(a)}$ is normal and all elements of a functional calculuscommutate.^[9]

Theholomorphic functional calculus isextended by the continuous functional calculus in an unambiguousway.^[10] Therefore, for polynomials${\displaystyle p(z,\overline{z})}$ the continuous functional calculus corresponds to the natural functional calculus for polynomials:${\mathrm{\Phi }}_a(p(z,\overline{z}))=p(a,a^{\ast })=\sum _{k,l=0}^N{c}_{k,l}{a}^k(a^{\ast }{)}^l$ for all$p(z,\overline{z})=\sum _{k,l=0}^N{c}_{k,l}{z}^k{\overline{z}}^l$ with${\displaystyle c_{k,l}\in \mathbb{C}}$.^[3]

For a sequence of functions${\displaystyle f_n\in C(\sigma (a))}$ that converges uniformly on${\displaystyle \sigma (a)}$ to a function${\displaystyle f\in C(\sigma (a))}$,${\displaystyle f_n(a)}$ converges to${\displaystyle f(a)}$.^[11] For apower series$f(z)=\sum _{n=0}^{\mathrm{\infty }}{c}_n{z}^n$, which convergesabsolutelyuniformly on${\displaystyle \sigma (a)}$, therefore$f(a)=\sum _{n=0}^{\mathrm{\infty }}{c}_n{a}^n$holds.^[12]

If${\displaystyle f\in \mathcal{C}(\sigma (a))}$ and${\displaystyle g\in \mathcal{C}(\sigma (f(a)))}$, then${\displaystyle (g\circ f)(a)=g(f(a))}$ holds for theircomposition.^[5] If${\displaystyle a,b\in {\mathcal{A}}_N}$ are two normal elements with${\displaystyle f(a)=f(b)}$ and${\displaystyle g}$ is theinverse function of${\displaystyle f}$ on both${\displaystyle \sigma (a)}$ and${\displaystyle \sigma (b)}$, then${\displaystyle a=b}$, since${\displaystyle a=(f\circ g)(a)=f(g(a))=f(g(b))=(f\circ g)(b)=b}$.^[13]

Thespectral mapping theorem applies:${\displaystyle \sigma (f(a))=f(\sigma (a))}$ for all${\displaystyle f\in C(\sigma (a))}$.^[7]

If${\displaystyle ab=ba}$ holds for${\displaystyle b\in \mathcal{A}}$, then${\displaystyle f(a)b=bf(a)}$ also holds for all${\displaystyle f\in C(\sigma (a))}$, i.e. if${\displaystyle b}$ commutates with${\displaystyle a}$, then also with the corresponding elements of the continuous functional calculus${\displaystyle f(a)}$.^[14]

Let${\displaystyle \mathrm{\Psi }:\mathcal{A}\to \mathcal{B}}$ be an unital *-homomorphism between C*-algebras${\displaystyle \mathcal{A}}$ and${\displaystyle \mathcal{B}}$. Then${\displaystyle \mathrm{\Psi }}$ commutates with the continuous functional calculus. The following holds:${\displaystyle \mathrm{\Psi }(f(a))=f(\mathrm{\Psi }(a))}$ for all${\displaystyle f\in C(\sigma (a))}$. In particular, the continuous functional calculus commutates with the Gelfandrepresentation.^[4]

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

• ${\displaystyle f(a)}$ isinvertible if and only if${\displaystyle f}$has no zero on${\displaystyle \sigma (a)}$.^[16] Then$f(a{)}^{-1}=\frac{1}{f}(a)$holds.^[17]

• ${\displaystyle f(a)}$ isself-adjoint if and only if${\displaystyle f}$ isreal-valued, i.e.${\displaystyle f(\sigma (a))\subseteq \mathbb{R}}$.
• ${\displaystyle f(a)}$ ispositive (${\displaystyle f(a)\ge 0}$) if and only if${\displaystyle f\ge 0}$, i.e.${\displaystyle f(\sigma (a))\subseteq [0,\mathrm{\infty })}$.
• ${\displaystyle f(a)}$ isunitary if all values of${\displaystyle f}$ lie in thecircle group, i.e.${\displaystyle f(\sigma (a))\subseteq \mathbb{T}=\{\lambda \in \mathbb{C}\mid \Vert \lambda \Vert =1\}}$.
• ${\displaystyle f(a)}$ is aprojection if${\displaystyle f}$ only takes on the values${\displaystyle 0}$ and${\displaystyle 1}$, i.e.${\displaystyle f(\sigma (a))\subseteq \{0,1\}}$.

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

In the special case that${\displaystyle \mathcal{A}}$ is the C*-algebra of bounded operators${\displaystyle \mathcal{B}(H)}$ for a Hilbert space${\displaystyle H}$,eigenvectors${\displaystyle v\in H}$ for the eigenvalue${\displaystyle \lambda \in \sigma (T)}$ of a normal operator${\displaystyle T\in \mathcal{B}(H)}$ are also eigenvectors for the eigenvalue${\displaystyle f(\lambda )\in \sigma (f(T))}$ of the operator${\displaystyle f(T)}$. If${\displaystyle Tv=\lambda v}$, then${\displaystyle f(T)v=f(\lambda )v}$ also holds for all${\displaystyle f\in \sigma (T)}$.^[18]

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

Let${\displaystyle \mathcal{A}}$ be a C*-algebra and${\displaystyle a\in {\mathcal{A}}_N}$ a normal element. Then the following applies to the spectrum${\displaystyle \sigma (a)}$:^[15]

• ${\displaystyle a}$ is self-adjoint if and only if${\displaystyle \sigma (a)\subseteq \mathbb{R}}$.
• ${\displaystyle a}$ is unitary if and only if${\displaystyle \sigma (a)\subseteq \mathbb{T}=\{\lambda \in \mathbb{C}\mid \Vert \lambda \Vert =1\}}$.
• ${\displaystyle a}$ is a projection if and only if${\displaystyle \sigma (a)\subseteq \{0,1\}}$.

Proof.^[3] The continuous functional calculus${\displaystyle {\mathrm{\Phi }}_a}$ for the normal element${\displaystyle a\in \mathcal{A}}$ is a *-homomorphism with${\displaystyle {\mathrm{\Phi }}_a(\mathrm{Id})=a}$ and thus${\displaystyle a}$ is self-adjoint/unitary/a projection if${\displaystyle \mathrm{Id}\in C(\sigma (a))}$ is also self-adjoint/unitary/a projection. Exactly then${\displaystyle \mathrm{Id}}$ is self-adjoint if${\displaystyle z=\text{Id}(z)=\overline{\text{Id}}(z)=\overline{z}}$ holds for all${\displaystyle z\in \sigma (a)}$, i.e. if${\displaystyle \sigma (a)}$ is real. Exactly then${\displaystyle \text{Id}}$ is unitary if${\displaystyle 1=\text{Id}(z)\overline{\mathrm{Id}}(z)=z\overline{z}=|z{|}^2}$ holds for all${\displaystyle z\in \sigma (a)}$, therefore${\displaystyle \sigma (a)\subseteq \{\lambda \in \mathbb{C}|\Vert \lambda \Vert =1\}}$. Exactly then${\displaystyle \text{Id}}$ is a projection if and only if${\displaystyle (\mathrm{Id}(z){)}^2=\mathrm{Id}}(z)=\overline{\mathrm{Id}(z)}$, that is${\displaystyle z^2=z=\overline{z}}$ for all${\displaystyle z\in \sigma (a)}$, i.e.${\displaystyle \sigma (a)\subseteq \{0,1\}}$

Let${\displaystyle a}$ be a positive element of a C*-algebra${\displaystyle \mathcal{A}}$. Then for every${\displaystyle n\in \mathbb{N}}$ there exists a uniquely determined positive element${\displaystyle b\in {\mathcal{A}}_{+}}$ with${\displaystyle b^n=a}$, i.e. a unique${\displaystyle n}$-throot.^[19]

Proof. For each${\displaystyle n\in \mathbb{N}}$, the root function${\displaystyle f_n:{\mathbb{R}}_0^{+}\to {\mathbb{R}}_0^{+},x\mapsto \sqrt[n]{x}}$ is a continuous function on${\displaystyle \sigma (a)\subseteq {\mathbb{R}}_0^{+}}$. If${\displaystyle b:=f_n(a)}$ is defined using the continuous functional calculus, then${\displaystyle b^n=(f_n(a){)}^n=(f_n^n)(a)={\mathrm{Id}}_{\sigma (a)}(a)=a}$ follows from the properties of the calculus. From the spectral mapping theorem follows${\displaystyle \sigma (b)=\sigma (f_n(a))=f_n(\sigma (a))\subseteq [0,\mathrm{\infty })}$, i.e.${\displaystyle b}$ ispositive.^[19] If${\displaystyle c\in {\mathcal{A}}_{+}}$ is another positive element with${\displaystyle c^n=a=b^n}$, then${\displaystyle c=f_n(c^n)=f_n(b^n)=b}$ holds, as the root function on the positive real numbers is an inverse function to the function${\displaystyle z\mapsto z^n}$.^[13]

If${\displaystyle a\in {\mathcal{A}}_{sa}}$ is a self-adjoint element, then at least for every odd${\displaystyle n\in \mathbb{N}}$ there is a uniquely determined self-adjoint element${\displaystyle b\in {\mathcal{A}}_{sa}}$ with${\displaystyle b^n=a}$.^[20]

Similarly, for a positive element${\displaystyle a}$ of a C*-algebra${\displaystyle \mathcal{A}}$, each${\displaystyle \alpha \ge 0}$ defines a uniquely determined positive element${\displaystyle a^{\alpha }}$ of${\displaystyle C^{\ast }(a)}$, such that${\displaystyle a^{\alpha }{a}^{\beta }=a^{\alpha +\beta }}$ holds for all${\displaystyle \alpha ,\beta \ge 0}$. If${\displaystyle a}$ is invertible, this can also be extended to negative values of${\displaystyle \alpha }$.^[19]

If${\displaystyle a\in \mathcal{A}}$, then the element${\displaystyle a^{\ast }a}$ is positive, so that the absolute value can be defined by the continuous functional calculus${\displaystyle |a|=\sqrt{a^{\ast }a}}$, since it is continuous on the positive realnumbers.^[21]

Let${\displaystyle a}$ be a self-adjoint element of a C*-algebra${\displaystyle \mathcal{A}}$, then there exist positive elements${\displaystyle a_{+},a_{-}\in {\mathcal{A}}_{+}}$, such that${\displaystyle a=a_{+}-a_{-}}$ with${\displaystyle a_{+}{a}_{-}=a_{-}{a}_{+}=0}$ holds. The elements${\displaystyle a_{+}}$ and${\displaystyle a_{-}}$ are also referred to as thepositive and negative parts.^[22] In addition,${\displaystyle |a|=a_{+}+a_{-}}$holds.^[23]

Proof. The functions${\displaystyle f_{+}(z)=max(z,0)}$ and${\displaystyle f_{-}(z)=-min(z,0)}$ are continuous functions on${\displaystyle \sigma (a)\subseteq \mathbb{R}}$ with${\displaystyle \mathrm{Id}(z)=z=f_{+}(z)-f_{-}(z)}$ and${\displaystyle f_{+}(z)f_{-}(z)=f_{-}(z)f_{+}(z)=0}$. Put${\displaystyle a_{+}=f_{+}(a)}$ and${\displaystyle a_{-}=f_{-}(a)}$. According to the spectral mapping theorem,${\displaystyle a_{+}}$ and${\displaystyle a_{-}}$ are positive elements for which${\displaystyle a=\mathrm{Id}(a)=(f_{+}-f_{-})(a)=f_{+}(a)-f_{-}(a)=a_{+}-a_{-}}$ and${\displaystyle a_{+}{a}_{-}=f_{+}(a)f_{-}(a)=(f_{+}{f}_{-})(a)=0=(f_{-}{f}_{+})(a)=f_{-}(a)f_{+}(a)=a_{-}{a}_{+}}$holds.^[22] Furthermore,$f_{+}(z)+f_{-}(z)=|z|=\sqrt{z^{\ast }z}=\sqrt{z^2}$, such that$a_{+}+a_{-}=f_{+}(a)+f_{-}(a)=|a|=\sqrt{a^{\ast }a}=\sqrt{a^2}$ holds.^[23]

If${\displaystyle a}$ is a self-adjoint element of a C*-algebra${\displaystyle \mathcal{A}}$ with unit element${\displaystyle e}$, then${\displaystyle u={\mathrm{e}}^{\mathrm{i}a}}$ is unitary, where${\displaystyle \mathrm{i}}$ denotes theimaginary unit. Conversely, if${\displaystyle u\in {\mathcal{A}}_U}$ is an unitary element, with the restriction that the spectrum is aproper subset of the unit circle, i.e.${\displaystyle \sigma (u)⊊\mathbb{T}}$, there exists a self-adjoint element${\displaystyle a\in {\mathcal{A}}_{sa}}$ with${\displaystyle u={\mathrm{e}}^{\mathrm{i}a}}$.^[24]

Proof.^[24] It is${\displaystyle u=f(a)}$ with${\displaystyle f:\mathbb{R}\to \mathbb{C},x\mapsto {\mathrm{e}}^{\mathrm{i}x}}$, since${\displaystyle a}$ is self-adjoint, it follows that${\displaystyle \sigma (a)\subset \mathbb{R}}$, i.e.${\displaystyle f}$ is a function on the spectrum of${\displaystyle a}$. Since${\displaystyle f\cdot \overline{f}=\overline{f}\cdot f=1}$, using the functional calculus${\displaystyle u{u}^{\ast }=u^{\ast }u=e}$ follows, i.e.${\displaystyle u}$ is unitary. Since for the other statement there is a${\displaystyle z_0\in \mathbb{T}}$, such that${\displaystyle \sigma (u)\subseteq \{{\mathrm{e}}^{\mathrm{i}z}\mid z_0\le z\le z_0+2\pi \}}$ the function${\displaystyle f({\mathrm{e}}^{\mathrm{i}z})=z}$ is a real-valued continuous function on the spectrum${\displaystyle \sigma (u)}$ for${\displaystyle z_0\le z\le z_0+2\pi }$, such that${\displaystyle a=f(u)}$ is a self-adjoint element that satisfies${\displaystyle {\mathrm{e}}^{\mathrm{i}a}={\mathrm{e}}^{\mathrm{i}f(u)}=u}$.

Let${\displaystyle \mathcal{A}}$ be an unital C*-algebra and${\displaystyle a\in {\mathcal{A}}_N}$ a normal element. Let the spectrum consist of${\displaystyle n}$ pairwisedisjointclosed subsets${\displaystyle {\sigma }_k\subset \mathbb{C}}$ for all${\displaystyle 1\le k\le n}$, i.e.${\displaystyle \sigma (a)={\sigma }_1\bigsqcup \cdots \bigsqcup {\sigma }_n}$. Then there exist projections${\displaystyle p_1,\dots ,p_n\in \mathcal{A}}$ that have the following properties for all${\displaystyle 1\le j,k\le n}$:^[25]

• For the spectrum,${\displaystyle \sigma (p_k)={\sigma }_k}$ holds.
• The projections commutate with${\displaystyle a}$, i.e.${\displaystyle p_{k}a=a{p}_k}$.
• The projections areorthogonal, i.e.${\displaystyle p_j{p}_k={\delta }_{jk}{p}_k}$.
• The sum of the projections is the unit element, i.e.$\sum _{k=1}^n{p}_k=e$.

In particular, there is a decomposition$a=\sum _{k=1}^n{a}_k$ for which${\displaystyle \sigma (a_k)={\sigma }_k}$ holds for all${\displaystyle 1\le k\le n}$.

Proof.^[25] Since all${\displaystyle {\sigma }_k}$ are closed, thecharacteristic functions${\displaystyle {\chi }_{{\sigma }_k}}$ are continuous on${\displaystyle \sigma (a)}$. Now let${\displaystyle p_k:={\chi }_{{\sigma }_k}(a)}$ be defined using the continuous functional. As the${\displaystyle {\sigma }_k}$ are pairwise disjoint,${\displaystyle {\chi }_{{\sigma }_j}{\chi }_{{\sigma }_k}={\delta }_{jk}{\chi }_{{\sigma }_k}}$ and$\sum _{k=1}^n{\chi }_{{\sigma }_k}={\chi }_{{\cup }_{k=1}^n{\sigma }_k}={\chi }_{\sigma (a)}=\mathbf{\text{1}}$ holds and thus the${\displaystyle p_k}$ satisfy the claimed properties, as can be seen from the properties of the continuous functional equation. For the last statement, let${\displaystyle a_k=a{p}_k=\mathrm{Id}(a)\cdot {\chi }_{{\sigma }_k}(a)=(\mathrm{Id}\cdot {\chi }_{{\sigma }_k})(a)}$.

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