continuous functional calculus

Let $B(H)$ be the algebra of bounded operators over a complex Hilbert space $H$. Let $T\in B(H)$ be a normal operator.

The continuous functional calculus is a functional calculus which enables the expression

$$f(T)$$

to make sense as a bounded operator in $H$, for continuous functions $f$.

More generally, when $𝒜$ is a $C^{*}$-algebra (http://planetmath.org/CAlgebra) with identity element $e$, and $x$ is a normal element of $𝒜$, the continuous functional calculus allows one to define $f(x)\in 𝒜$ when $f$ is a continuous function.

More precisely, if $\sigma (x)$ denotes the spectrum of $x$ and $C(\sigma (x))$ denotes the $C^{*}$-algebra of complex valued continuous functions on $\sigma (x)$, we will define a continuous homomorphism

$C(\sigma (x))⟶𝒜$

$f\mapsto f(x)$

that the functional calculus (http://planetmath.org/FunctionalCalculus) .

There are several reasons to require the continuity of $f$ on the spectrum $\sigma (x)$.

For example, suppose ${\lambda }_0\in \sigma (x)$. The function $f(\lambda )=\frac{1}{\lambda -{\lambda }_0}$ is clearly not continuous in ${\lambda }_0$. By the functional calculus we would obtain

$$f(x)=\frac{1}{x-{\lambda }_{0}e}={(x-{\lambda }_{0}e)}^{-1}$$

but $x-{\lambda }_{0}e$ is not invertible since ${\lambda }_0\in \sigma (x)$.

The abstraction towards $C^{*}$-algebras is almost . Indeed, $C^{*}$-algebras are the appropriate where to and prove the continuous functional calculus. The conclusions towards $B(H)$ then follow as a particular case.

1 Preliminary construction

Let $𝒜$ be a unital $C^{*}$-algebra and $x$ a normal element in $𝒜$. Let $ℬ\subseteq 𝒜$ be the $C^{*}$-subalgebra generated by $x$ and the identity $e$ of $𝒜$.

Thus, $ℬ$ is the norm closure of the algebra generated by $x$, $x^{*}$ and $e$.

Moreover, since $x$ is , $x^{*}x=x{x}^{*}$, it follows that $ℬ$ is commutative and $ℬ$ consists of those elements $y\in 𝒜$ that can be approximated by polynomials $p(x,x^{*})$ in $x$ and $x^{*}$.

Recall the following facts:

• Since $ℬ$ is a commutative unital $C^{*}$-algebra, the set $△$ of multiplicative linear functionals on $ℬ$ is a compact Hausdorff space.

• Let $C(△)$ denote the $C^{*}$-algebra of complex valued continuous functions on $△$. The Gelfand transform $G:ℬ⟶C(△)$ is a $C^{*}$-isomorphism.

The following result is perhaps the for the definition of the continuous functional calculus.

Theorem 1 - $\mathrm{△}$ and $\sigma \mathit{}\mathrm{(}x\mathrm{)}$ are homeomorphic topological spaces. Moreover, the mapping $S\mathrm{:}\mathrm{△}\mathrm{\to }\sigma \mathrm{(}x\mathrm{)}$ defined by

$S(\varphi ):=\varphi (x)$

is such an homeomorphism.

: We need to check that $S$ is well defined, i.e. $\varphi (x)\in \sigma (x)$ for all $\varphi \in △$.

From the identity

$$\varphi (x-\varphi (x)e)=\varphi (x)-\varphi (x)\varphi (e)=\varphi (x)-\varphi (x)=0$$

follows that $x-\varphi (x)e$ cannot be invertible in $ℬ$ (recall that $\varphi$ is a multiplicative linear functional on $ℬ$).

Thus, $\varphi (x)\in {\sigma }_{ℬ}(x)$. By the spectral invariance theorem, we see that $\varphi (x)\in \sigma (x)={\sigma }_{ℬ}(x)$, and so $S$ is well defined.

• $S$ is continuous - Suppose ${\varphi }_{\alpha }$ is a net in $△$ such that ${\varphi }_{\alpha }⟶\varphi$. Recall that the topology in $△$ is the weak-* topology, so ${\varphi }_{\alpha }(x)⟶\varphi (x)$.

  Thus, $S({\varphi }_{\alpha })⟶S(\varphi )$ and so $S$ is continuous.

• $S$ is - Suppose $S({\varphi }_1)=S({\varphi }_2)$. Then, ${\varphi }_1(x)={\varphi }_2(x)$. Since

  $${\varphi }_i(x^{*})=G(x^{*})({\varphi }_i)=\overline{G(x)({\varphi }_i)}=\overline{{\varphi }_i(x)}$$

  we must also have ${\varphi }_1(x^{*})={\varphi }_2(x^{*})$.

  This clearly implies that

  $${\varphi }_1(p(x,x^{*}))={\varphi }_2(p(x,x^{*}))$$

  for every polynomial in two variables $p$.

  Recall that the ”polynomials” $p(x,x^{*})$ are dense in $ℬ$. So we must have ${\varphi }_1(y)={\varphi }_2(y)$ for every $y\in ℬ$, i.e. ${\varphi }_1={\varphi }_2$.

• $S$ is - Let $\lambda \in \sigma (x)={\sigma }_{ℬ}(x)$. Then $x-\lambda e$ is not invertible.

  Since $ℬ$ is commutative, $x-\lambda e$ is contained in a maximal ideal $ℳ$.

  As $ℳ$ is maximal ideal, the quotient $ℬ/ℳ$ is a division algebra, and so by the Gelfand-Mazur theorem, $ℬ/ℳ$ must ne isomorphic to $ℂ$.

  Therefore the quotient homomorphism

  $$\varphi :ℬ⟶ℬ/ℳ=ℂ$$

  is a multiplicative linear functional such that $\varphi (x-\lambda e)=0$, i.e. $\varphi (x)=\lambda$, i.e. $S(\varphi )=\lambda$.

  Therefore, $S$ is surjective.

Since $S$ is a continuous bijective function from the compact Hausdorff space $△$ to $\sigma (x)$, it follows that it must be a homeomorphism. $\mathrm{\square }$

2 Definition of the continuous functional calculus

If $S$ is the homeomorphism between $△$ and $\sigma (x)$ defined as above, then the mapping $f\mapsto f\circ S$ is a *-isomorphism between the algebras $C(\sigma (x))$ and $C(△)$. Since the Gelfand transform $G:ℬ⟶C(△)$ is a also a *-isomorphism, we obtain a *-isomorphism

$$\mathrm{\Gamma }:C(\sigma (x))⟶ℬ$$

by setting $\mathrm{\Gamma }(f):=G^{-1}(f\circ S)$.

Definition - Suppose $x$ is a normal element in a unital $C^{\mathrm{*}}$-algebra $\mathrm{A}$. For every $f\mathrm{\in }C\mathit{}\mathrm{(}\sigma \mathit{}\mathrm{(}x\mathrm{)}\mathrm{)}$ we define

$$f(x):=\mathrm{\Gamma }(f)\in ℬ\subseteq 𝒜$$

The mapping $\mathrm{\Gamma }$, such that $f\mathrm{\mapsto }f\mathit{}\mathrm{(}x\mathrm{)}$, is called the continuous functional calculus for $x$.

We now prove the functional calculus (http://planetmath.org/FunctionalCalculus) for the continuous functional calculus and show its uniqueness:

Theorem 2 - Let $\mathrm{A}$ be a unital $C^{\mathrm{*}}$-algebra, $x\mathrm{\in }\mathrm{A}$ a normal element and $\mathrm{id}$ the identity function in $\mathrm{C}$. The continuous functional calculus for $x$ is the unique unital *-homomorphism between $C\mathit{}\mathrm{(}\sigma \mathit{}\mathrm{(}x\mathrm{)}\mathrm{)}$ and $\mathrm{A}$ which sends $\mathrm{id}$ to $x$. In particular, for every polynomial $p$ in $\mathrm{C}$ of the form $p\mathit{}\mathrm{(}\lambda \mathrm{)}\mathrm{:=}\mathrm{\sum }{c}_{n\mathrm{,}m}\mathit{}{\lambda }^n\mathit{}{\overline{\lambda }}^m$, we have $p\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}\mathrm{\sum }{c}_{n\mathrm{,}m}\mathit{}{x}^n\mathit{}{\mathrm{(}{x}^{\mathrm{*}}\mathrm{)}}^m$.

: We have seen that the continuous functional calculus $\mathrm{\Gamma }$ for $x$ is a *-homomorphism between $C(\sigma (x))$ and $𝒜$. Recall that $\mathrm{\Gamma }$ was defined by $\mathrm{\Gamma }(f):=G^{-1}(f\circ S)$. It is clear by the definition that $\mathrm{\Gamma }$ is unital. Also, $G\circ \mathrm{\Gamma }(f)=f\circ S$ for every $f\in C(△)$. Taking the identity function $\mathrm{id}$ we obtain that for every $\varphi \in △$

$G\circ \mathrm{\Gamma }(\mathrm{id})(\varphi )$ $=$ $\mathrm{id}(S(\varphi ))$

$=$ $\varphi (x)$

$=$ $G(x)(\varphi )$

Since the Gelfand transform is a *-isomorphism, we must have $\mathrm{\Gamma }(\mathrm{id})=x$.

Now, let $p:ℂ\to ℂ$ be a polynomial of the form $p(\lambda ):=\sum c_{n,m}{\lambda }^n{\overline{\lambda }}^m$. Notice that $p=\sum c_{n,m}{\mathrm{id}}^n{\overline{\mathrm{id}}}^m$. If $F$ is any unital *-homomorphism such that $F(\mathrm{id})=x$, then one must have $F(p)=\sum c_{n,m}{x}^n{(x^{*})}^m$. Thus all such unital *-homomorphisms coincide on the subspace of polynomials of the above form. By the Stone-Weierstrass theorem (http://planetmath.org/StoneWeierstrassTheoremComplexVersion), this subspace is dense in $C(\sigma (x))$. Thus, all such unital *-homomorphisms coincide in $C(\sigma (x))$, and uniqueness is proven. $\mathrm{\square }$

3 Properties

• The spectral mapping theorem assures that for $f\in C(\sigma (x))$

  $$\sigma (f(x))=f(\sigma (x))$$

• When $\sigma (x)\subset {ℝ}^{+}$ the continuous functional calculus assures the existence of a square root $\sqrt{x}$ of $x$, since $\sqrt{\lambda }$ is defined and continuous on $\lambda \in \sigma (x)$.

Title	continuous functional calculus
Canonical name	ContinuousFunctionalCalculus
Date of creation	2013-03-22 17:30:02
Last modified on	2013-03-22 17:30:02
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	11
Author	asteroid (17536)
Entry type	Feature
Classification	msc 47A60
Classification	msc 46L05
Classification	msc 46H30
Related topic	FunctionalCalculus
Related topic	PolynomialFunctionalCalculus
Related topic	BorelFunctionalCalculus
Defines	continuous functions of normal operators