Continuous functional calculus

Theorem

continuous functional calculus—Let ${\displaystyle a}$ be a normal element of the C*-algebra ${\displaystyle \mathcal{A}}$ with unit element ${\displaystyle e}$ and let ${\displaystyle C(\sigma (a))}$ be the commutative C*-algebra of continuous functions on ${\displaystyle \sigma (a)}$, the spectrum of ${\displaystyle a}$. Then there exists exactly one *-homomorphism ${\displaystyle {\mathrm{\Phi }}_a:C(\sigma (a))\to \mathcal{A}}$ with ${\displaystyle {\mathrm{\Phi }}_a(\mathbf{1})=e}$ for ${\displaystyle \mathbf{1}(z)=1}$ and ${\displaystyle {\mathrm{\Phi }}_a({\mathrm{Id}}_{\sigma (a)})=a}$ for the identity.

The mapping ${\displaystyle {\mathrm{\Phi }}_a}$ is called the continuous functional calculus of the normal element ${\displaystyle a}$. Usually it is suggestively set ${\displaystyle f(a):={\mathrm{\Phi }}_a(f)}$.

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

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 ${\displaystyle {\mathcal{A}}_1}$. 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 ${\displaystyle f(a)\in \mathcal{A}\subset {\mathcal{A}}_1}$.

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 ${\displaystyle \mathcal{A}=C^{\ast }(a,e)}$. The actual construction is almost immediate from the Gelfand representation: it suffices to assume ${\displaystyle \mathcal{A}}$ is the C*-algebra of continuous functions on some compact space ${\displaystyle X}$ and define ${\displaystyle {\mathrm{\Phi }}_a(f)=f\circ x}$.
• 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 a dense subalgebra of ${\displaystyle C(\sigma (a))}$ by the Stone-Weierstrass theorem. Thus ${\displaystyle {\mathrm{\Phi }}_a}$ is unique.

In functional 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 ${\displaystyle \mathcal{B}(H)}$ of bounded operators on a Hilbert space ${\displaystyle H}$. 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.

Further properties of the continuous functional calculus

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

• ${\displaystyle \Vert {\mathrm{\Phi }}_a(f)\Vert ={\Vert f\Vert }_{\sigma (a)}}$ 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 calculus commutate.

The holomorphic functional calculus is extended by the continuous functional calculus in an unambiguous way. 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 ${\displaystyle c_{k,l}\in \mathbb{C}}$.

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 ${\displaystyle f(a)}$. For a power series $f(z)=\sum _{n=0}^{\mathrm{\infty }}{c}_n{z}^n$, which converges absolutely uniformly on ${\displaystyle \sigma (a)}$, therefore $f(a)=\sum _{n=0}^{\mathrm{\infty }}{c}_n{a}^n$ holds.

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 their composition. If ${\displaystyle a,b\in {\mathcal{A}}_N}$ are two normal elements with ${\displaystyle f(a)=f(b)}$ and ${\displaystyle g}$ is the inverse function of ${\displaystyle f}$ on both ${\displaystyle \sigma (a)}$ and ${\displaystyle \sigma (b)}$, then ${\displaystyle a=b}$, since ${\displaystyle a=(f\circ g)(a)=f(g(a))=f(g(b))=(f\circ g)(b)=b}$.

The spectral mapping theorem applies: ${\displaystyle \sigma (f(a))=f(\sigma (a))}$ for all ${\displaystyle f\in C(\sigma (a))}$.

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 ${\displaystyle f(a)}$.

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 Gelfand representation.

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

• ${\displaystyle f(a)}$ is invertible if and only if ${\displaystyle f}$ has no zero on ${\displaystyle \sigma (a)}$. Then $f(a{)}^{-1}=\frac{1}{f}(a)$ holds.
• ${\displaystyle f(a)}$ is self-adjoint if and only if ${\displaystyle f}$ is real-valued, i.e. ${\displaystyle f(\sigma (a))\subseteq \mathbb{R}}$.
• ${\displaystyle f(a)}$ is positive (${\displaystyle f(a)\ge 0}$) if and only if ${\displaystyle f\ge 0}$, i.e. ${\displaystyle f(\sigma (a))\subseteq [0,\mathrm{\infty })}$.
• ${\displaystyle f(a)}$ is unitary if all values of ${\displaystyle f}$ lie in the circle group, i.e. ${\displaystyle f(\sigma (a))\subseteq \mathbb{T}=\{\lambda \in \mathbb{C}\mid \Vert \lambda \Vert =1\}}$.
• ${\displaystyle f(a)}$ is a projection 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 ${\displaystyle f\in \sigma (T)}$.

Applications

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

Spectrum

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 ${\displaystyle \sigma (a)}$:

• ${\displaystyle a}$ 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. 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. ${\displaystyle \sigma (a)\subseteq \{0,1\}}$

Roots

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 ${\displaystyle n}$-th root.

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. ${\displaystyle b}$ is positive. 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 ${\displaystyle z\mapsto z^n}$.

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 ${\displaystyle b^n=a}$.

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 ${\displaystyle \alpha }$.

Absolute value

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 real numbers.

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 the positive and negative parts. In addition, ${\displaystyle |a|=a_{+}+a_{-}}$ holds.

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. 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.

Unitary elements

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 the imaginary unit. Conversely, if ${\displaystyle u\in {\mathcal{A}}_U}$ is a unitary element, with the restriction that the spectrum is a proper 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 ${\displaystyle u={\mathrm{e}}^{\mathrm{i}a}}$.

Proof. 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 ${\displaystyle {\mathrm{e}}^{\mathrm{i}a}={\mathrm{e}}^{\mathrm{i}f(u)}=u}$.

Spectral decomposition theorem

Let ${\displaystyle \mathcal{A}}$ be an unital C*-algebra and ${\displaystyle a\in {\mathcal{A}}_N}$ a normal element. Let the spectrum consist of ${\displaystyle n}$ pairwise disjoint closed 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 ${\displaystyle 1\le j,k\le n}$:

• 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 are orthogonal, 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. Since all ${\displaystyle {\sigma }_k}$ are closed, the characteristic 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 ${\displaystyle a_k=a{p}_k=\mathrm{Id}(a)\cdot {\chi }_{{\sigma }_k}(a)=(\mathrm{Id}\cdot {\chi }_{{\sigma }_k})(a)}$.