Английская Википедия: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
If one wants to extend the natural functional calculus for polynomials on the spectrum <math>\sigma(a)</math> of an element <math>a</math> of a Banach algebra <math>\mathcal{A}</math> to a functional calculus for continuous functions <math>C(\sigma(a))</math> 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 Шаблон:Nowrap The continuous functions on <math>\sigma(a) \subset \C</math> are approximated by polynomials in <math>z</math> and <math>\overline{z}</math>, i.e. by polynomials of the form Шаблон:Nowrap Here, <math>\overline{z}</math> denotes the complex conjugation, which is an involution on the Шаблон:Nowrap To be able to insert <math>a</math> in place of <math>z</math> in this kind of polynomial, Banach *-algebras are considered, i.e. Banach algebras that also have an involution *, and <math>a^*</math> is inserted in place of Шаблон:Nowrap In order to obtain a homomorphism <math>{\mathbb C}[z,\overline{z}]\rightarrow\mathcal{A}</math>, a restriction to normal elements, i.e. elements with <math>a^*a = aa^*</math>, is necessary, as the polynomial ring <math>\C[z,\overline{z}]</math> is commutative. If <math>(p_n(z,\overline{z}))_n</math> is a sequence of polynomials that converges uniformly on <math>\sigma(a)</math> to a continuous function <math>f</math>, the convergence of the sequence <math>(p_n(a,a^*))_n</math> in <math>\mathcal{A}</math> to an element <math>f(a)</math> 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
Due to the *-homomorphism property, the following calculation rules apply to all functions <math>f,g \in C(\sigma(a))</math> and scalars <math>\lambda,\mu \in \C</math>:Шаблон:Sfn
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(linear) |
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(multiplicative) |
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(involutive) |
One can therefore imagine actually inserting the normal elements into continuous functions; the obvious algebraic operations behave as expected.
The requirement for an unit element is not a significant restriction. If necessary, an unit element can be adjoint and then operated in the enlarged C*-algebra Шаблон:Nowrap Then if <math>a \in \mathcal{A}</math> and <math>f \in C(\sigma (a))</math> with <math>f(0) = 0</math>, it follows that <math>0 \in \sigma (a)</math> and Шаблон:Nowrap
The existence and uniqueness of the continuous functional calculus are proven separately:
- Existence: Since the spectrum of <math>a</math> in the C*-subalgebra <math>C^*(a,e)</math> generated by <math>a</math> and <math>e</math> is the same as it is in <math>\mathcal{A}</math>, it suffices to show the statement for Шаблон:Nowrap The actual construction is almost immediate from the Gelfand representation: it suffices to assume <math>\mathcal{A}</math> is the C*-algebra of continuous functions on some compact space <math>X</math> and define Шаблон:Nowrap
- Uniqueness: Since <math>\Phi_a(\boldsymbol{1})</math> and <math>\Phi_a(\operatorname{Id}_{\sigma(a)})</math> are fixed, <math>\Phi_a</math> is already uniquely defined for all polynomials <math display="inline">p(z, \overline{z}) = \sum_{k,l=0}^N c_{k,l} z^k\overline{z}^l \; \left( c_{k,l} \in \C \right)</math>, since <math>\Phi_a</math> is a *-homomorphism. These form a dense subalgebra of <math>C(\sigma(a))</math> by the Stone-Weierstrass theorem. Thus <math>\Phi_a</math> is Шаблон:Nowrap
In functional analysis, the continuous functional calculus for a normal operator <math>T</math> is often of interest, i.e. the case where <math>\mathcal{A}</math> is the C*-algebra <math>\mathcal{B}(H)</math> of bounded operators on a Hilbert space Шаблон:Nowrap 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 Шаблон:Nowrap
Further properties of the continuous functional calculus
The continuous functional calculus <math>\Phi_a</math> is an isometric isomorphism into the C*-subalgebra <math>C^*(a,e)</math> generated by <math>a</math> and <math>e</math>, that is:Шаблон:Sfn
- <math>\left\| \Phi_a(f) \right\| = \left\| f \right\|_{\sigma(a)}</math> for all <math>f \in C(\sigma(a))</math>; <math>\Phi_a</math> is therefore continuous.
- <math>\Phi_a \left( C(\sigma(a)) \right) = C^*(a, e) \subseteq \mathcal{A}</math>
Since <math>a</math> is a normal element of <math>\mathcal{A}</math>, the C*-subalgebra generated by <math>a</math> and <math>e</math> is commutative. In particular, <math>f(a)</math> is normal and all elements of a functional calculus Шаблон:Nowrap
The holomorphic functional calculus is extended by the continuous functional calculus in an unambiguous Шаблон:Nowrap Therefore, for polynomials <math>p(z,\overline{z})</math> the continuous functional calculus corresponds to the natural functional calculus for polynomials: <math display="inline">\Phi_a(p(z, \overline{z})) = p(a, a^*) = \sum_{k,l=0}^N c_{k, l} a^k(a^*)^l</math> for all Шаблон:Nowrap
For a sequence of functions <math>f_n \in C(\sigma(a))</math> that converges uniformly on <math>\sigma(a)</math> to a function <math>f \in C(\sigma(a))</math>, <math>f_n(a)</math> converges to Шаблон:NowrapШаблон:Sfn For a power series <math display="inline">f(z) = \sum_{n=0}^\infty c_n z^n</math>, which converges absolutely uniformly on <math>\sigma(a)</math>, therefore <math display="inline">f(a) = \sum_{n=0}^\infty c_na^n</math> Шаблон:Nowrap
If <math>f \in \mathcal{C}(\sigma(a))</math> and <math>g\in \mathcal{ C}(\sigma(f(a)))</math>, then <math>(g \circ f)(a) = g(f(a))</math> holds for their Шаблон:Nowrap If <math>a,b \in \mathcal{A}_N</math> are two normal elements with <math>f(a) = f(b)</math> and <math>g</math> is the inverse function of <math>f</math> on both <math>\sigma(a)</math> and <math>\sigma(b)</math>, then <math>a = b</math>, since Шаблон:Nowrap
The spectral mapping theorem applies: <math>\sigma(f(a)) = f(\sigma(a))</math> for all Шаблон:Nowrap
If <math>ab = ba</math> holds for <math>b \in \mathcal{A}</math>, then <math>f(a)b = bf(a)</math> also holds for all <math>f \in C ( \sigma (a))</math>, i.e. if <math>b</math> commutates with <math>a</math>, then also with the corresponding elements of the continuous functional calculus Шаблон:Nowrap
Let <math>\Psi \colon \mathcal{A} \rightarrow \mathcal{B}</math> be an unital *-homomorphism between C*-algebras <math>\mathcal{A}</math> and Шаблон:Nowrap Then <math>\Psi</math> commutates with the continuous functional calculus. The following holds: <math>\Psi(f(a)) = f(\Psi(a))</math> for all Шаблон:Nowrap In particular, the continuous functional calculus commutates with the Gelfand Шаблон:Nowrap
With the spectral mapping theorem, functions with certain properties can be directly related to certain properties of elements of C*-algebras:Шаблон:Sfn
- <math>f(a)</math> is invertible if and only if <math>f</math> has no zero on Шаблон:Nowrap Then <math display="inline">f(a)^{-1} = \tfrac{1}{f} (a)</math> Шаблон:Nowrap
- <math>f(a)</math> is self-adjoint if and only if <math>f</math> is real-valued, i.e. Шаблон:Nowrap
- <math>f(a)</math> is positive (<math>f(a) \geq 0</math>) if and only if <math>f \geq 0</math>, i.e. Шаблон:Nowrap
- <math>f(a)</math> is unitary if all values of <math>f</math> lie in the circle group, i.e. Шаблон:Nowrap
- <math>f(a)</math> is a projection if <math>f</math> only takes on the values <math>0</math> and <math>1</math>, i.e. Шаблон:Nowrap
These are based on statements about the spectrum of certain elements, which are shown in the Applications section.
In the special case that <math>\mathcal{A}</math> is the C*-algebra of bounded operators <math>\mathcal{B}(H)</math> for a Hilbert space <math>H</math>, eigenvectors <math>v \in H</math> for the eigenvalue <math>\lambda \in \sigma(T)</math> of a normal operator <math>T \in \mathcal{B}(H)</math> are also eigenvectors for the eigenvalue <math>f(\lambda) \in \sigma(f(T))</math> of the operator Шаблон:Nowrap If <math>Tv = \lambda v</math>, then <math>f(T)v = f(\lambda)v</math> also holds for all Шаблон:Nowrap
Applications
The following applications are typical and very simple examples of the numerous applications of the continuous functional calculus:
Spectrum
Let <math>\mathcal{A}</math> be a C*-algebra and <math>a \in \mathcal{A}_N</math> a normal element. Then the following applies to the spectrum Шаблон:Nowrap
- <math>a</math> is self-adjoint if and only if Шаблон:Nowrap
- <math>a</math> is unitary if and only if Шаблон:Nowrap
- <math>a</math> is a projection if and only if Шаблон:Nowrap
Proof.Шаблон:Sfn The continuous functional calculus <math>\Phi_a</math> for the normal element <math>a \in \mathcal{A}</math> is a *-homomorphism with <math>\Phi_a (\operatorname{Id}) = a</math> and thus <math>a</math> is self-adjoint/unitary/a projection if <math>\operatorname{Id} \in C( \sigma(a))</math> is also self-adjoint/unitary/a projection. Exactly then <math>\operatorname{Id}</math> is self-adjoint if <math>z = \text{Id}(z) = \overline{\text{Id}}(z) = \overline{z}</math> holds for all <math>z \in \sigma(a)</math>, i.e. if <math>\sigma(a)</math> is real. Exactly then <math>\text{Id}</math> is unitary if <math>1 = \text{Id}(z) \overline{\operatorname{Id}}(z) = z \overline{z} = |z|^2</math> holds for all <math>z \in \sigma(a)</math>, therefore Шаблон:Nowrap Exactly then <math>\text{Id}</math> is a projection if and only if <math>(\operatorname{Id}(z))^2 = \operatorname{Id}}(z) = \overline{\operatorname{Id}(z)</math>, that is <math>z^2 = z = \overline{z}</math> for all <math>z \in \sigma(a)</math>, i.e. <math>\sigma(a) \subseteq \{ 0,1 \}</math>
Roots
Let <math>a</math> be a positive element of a C*-algebra Шаблон:Nowrap Then for every <math>n \in \mathbb{N}</math> there exists a uniquely determined positive element <math>b \in \mathcal{A}_+</math> with <math>b^n =a</math>, i.e. a unique <math>n</math>-th Шаблон:Nowrap
Proof. For each <math>n \in \mathbb{N}</math>, the root function <math>f_n \colon \R_0^+ \to \R_0^+, x \mapsto \sqrt[n]x</math> is a continuous function on Шаблон:Nowrap If <math>b \; \colon = f_n (a)</math> is defined using the continuous functional calculus, then <math>b^n = (f_n(a))^n = (f_n^n)(a) = \operatorname{Id}_{\sigma(a)}(a)=a</math> follows from the properties of the calculus. From the spectral mapping theorem follows <math>\sigma(b) = \sigma(f_n(a)) = f_n(\sigma(a)) \subseteq [0,\infty)</math>, i.e. <math>b</math> is Шаблон:Nowrap If <math>c \in \mathcal{A}_+</math> is another positive element with <math>c^n = a = b^n</math>, then <math>c = f_n (c^n) = f_n(b^n) = b</math> holds, as the root function on the positive real numbers is an inverse function to the function Шаблон:Nowrap
If <math>a \in \mathcal{A}_{sa}</math> is a self-adjoint element, then at least for every odd <math>n \in \N</math> there is a uniquely determined self-adjoint element <math>b \in \mathcal{A}_{sa}</math> with Шаблон:Nowrap
Similarly, for a positive element <math>a</math> of a C*-algebra <math>\mathcal{A}</math>, each <math>\alpha \geq 0</math> defines a uniquely determined positive element <math>a^\alpha</math> of <math>C^*(a)</math>, such that <math>a^\alpha a^\beta = a^{\alpha + \beta}</math> holds for all Шаблон:Nowrap If <math>a</math> is invertible, this can also be extended to negative values of Шаблон:Nowrap
Absolute value
If <math>a \in \mathcal{A}</math>, then the element <math>a^*a</math> is positive, so that the absolute value can be defined by the continuous functional calculus <math>|a| = \sqrt{a^*a}</math>, since it is continuous on the positive real Шаблон:Nowrap
Let <math>a</math> be a self-adjoint element of a C*-algebra <math>\mathcal{A}</math>, then there exist positive elements <math>a_+,a_- \in \mathcal{A}_+</math>, such that <math>a = a_+ - a_-</math> with <math>a_+ a_- = a_- a_+ = 0</math> holds. The elements <math>a_+</math> and <math>a_-</math> are also referred to as the Шаблон:Nowrap In addition, <math>|a| = a_+ + a_-</math> Шаблон:Nowrap
Proof. The functions <math>f_+(z) = \max(z,0)</math> and <math>f_-(z) = -\min(z, 0)</math> are continuous functions on <math>\sigma(a) \subseteq \R</math> with <math>\operatorname{Id} (z) = z = f_+(z) -f_-(z)</math> and Шаблон:Nowrap Put <math>a_+ = f_+(a)</math> and <math>a_- = f_-(a)</math>. According to the spectral mapping theorem, <math>a_+</math> and <math>a_-</math> are positive elements for which <math>a = \operatorname{Id}(a) = (f_+ - f_-) (a) = f_+(a) - f_-(a) = a_+ - a_-</math> and <math>a_+ a_- = f_+(a)f_-(a) = (f_+f_-)(a) = 0 = (f_-f_+)(a) = f_-(a)f_+(a) = a_- a_+</math> Шаблон:Nowrap Furthermore, <math display="inline">f_+(z) + f_-(z) = |z| = \sqrt{z^* z} = \sqrt{z^2}</math>, such that Шаблон:Nowrap
Unitary elements
If <math>a</math> is a self-adjoint element of a C*-algebra <math>\mathcal{A}</math> with unit element <math>e</math>, then <math>u = \mathrm{e}^{\mathrm{i} a}</math> is unitary, where <math>\mathrm{i}</math> denotes the imaginary unit. Conversely, if <math>u \in \mathcal{A}_U</math> is an unitary element, with the restriction that the spectrum is a proper subset of the unit circle, i.e. <math>\sigma(u) \subsetneq \mathbb{T}</math>, there exists a self-adjoint element <math>a \in \mathcal{A}_{sa}</math> with Шаблон:Nowrap
Proof.Шаблон:Sfn It is <math>u = f(a)</math> with <math>f \colon \R \to \C,\ x \mapsto \mathrm{e}^{\mathrm{i}x}</math>, since <math>a</math> is self-adjoint, it follows that <math>\sigma(a) \subset \R</math>, i.e. <math>f</math> is a function on the spectrum of Шаблон:Nowrap Since <math>f\cdot \overline{f} = \overline{f}\cdot f = 1</math>, using the functional calculus <math>uu^* = u^*u = e</math> follows, i.e. <math>u</math> is unitary. Since for the other statement there is a <math>z_0 \in \mathbb{T}</math>, such that <math>\sigma(u) \subseteq \{ \mathrm{e}^{\mathrm{i} z} \mid z_0 \leq z \leq z_0 + 2 \pi \}</math> the function <math>f(\mathrm{e}^{\mathrm{i} z}) = z</math> is a real-valued continuous function on the spectrum <math>\sigma(u)</math> for <math>z_0 \leq z \leq z_0 + 2 \pi</math>, such that <math>a = f(u)</math> is a self-adjoint element that satisfies Шаблон:Nowrap
Spectral decomposition theorem
Let <math>\mathcal{A}</math> be an unital C*-algebra and <math>a \in \mathcal{A}_N</math> a normal element. Let the spectrum consist of <math>n</math> pairwise disjoint closed subsets <math>\sigma_k \subset \C</math> for all <math>1 \leq k \leq n</math>, i.e. Шаблон:Nowrap Then there exist projections <math>p_1, \ldots, p_n \in \mathcal{A}</math> that have the following properties for all Шаблон:Nowrap
- For the spectrum, <math>\sigma(p_k) = \sigma_k</math> holds.
- The projections commutate with <math>a</math>, i.e. Шаблон:Nowrap
- The projections are orthogonal, i.e. Шаблон:Nowrap
- The sum of the projections is the unit element, i.e. Шаблон:Nowrap
In particular, there is a decomposition <math display="inline">a = \sum_{k=1}^n a_k</math> for which <math>\sigma(a_k) = \sigma_k</math> holds for all Шаблон:Nowrap
Proof.Шаблон:Sfn Since all <math>\sigma_k</math> are closed, the characteristic functions <math>\chi_{\sigma_k}</math> are continuous on Шаблон:Nowrap Now let <math>p_k := \chi_{\sigma_k} (a)</math> be defined using the continuous functional. As the <math>\sigma_k</math> are pairwise disjoint, <math>\chi_{\sigma_j} \chi_{\sigma_k} = \delta_{jk} \chi_{\sigma_k}</math> and <math display="inline">\sum_{k=1}^n \chi_{\sigma_k} = \chi_{\cup_{k=1}^n \sigma_k} = \chi_{\sigma(a)} = \textbf{1}</math> holds and thus the <math>p_k</math> satisfy the claimed properties, as can be seen from the properties of the continuous functional equation. For the last statement, let Шаблон:Nowrap
Notes
References
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book English translation of Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
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