🎵C*-algebras Unit 9 – Spectral Theory in C*–algebras

Key Concepts and Definitions

• C*-algebra a complex Banach algebra with an involution satisfying the C*-identity $\|a^*a\| = \|a\|^2$
• Spectrum of an element $a$ in a C*-algebra $A$, denoted by $\sigma(a)$, the set of all $\lambda \in \mathbb{C}$ such that $a - \lambda 1$ is not invertible in $A$
• Resolvent set of an element $a$ in a C*-algebra $A$, the complement of the spectrum, i.e., all $\lambda \in \mathbb{C}$ such that $a - \lambda 1$ is invertible in $A$
  • Resolvent function $R(\lambda, a) = (a - \lambda 1)^{-1}$ for $\lambda$ in the resolvent set of $a$
• Point spectrum of an element $a$, the set of all eigenvalues of $a$, i.e., all $\lambda \in \mathbb{C}$ such that $a - \lambda 1$ is not injective
• Continuous spectrum of an element $a$, the set of all $\lambda \in \sigma(a)$ that are not eigenvalues and for which $a - \lambda 1$ is not surjective
• Residual spectrum of an element $a$, the set of all $\lambda \in \sigma(a)$ that are not eigenvalues and for which $a - \lambda 1$ is not injective
• Spectral radius of an element $a$, denoted by $r(a)$, the radius of the smallest closed disk centered at the origin containing $\sigma(a)$, i.e., $r(a) = \sup\{|\lambda| : \lambda \in \sigma(a)\}$

Fundamental Theorems

• Spectral Theorem for Normal Elements if $a$ is a normal element in a C*-algebra $A$, then there exists a unique *-homomorphism $\Phi: C(\sigma(a)) \to C^*(a)$ such that $\Phi(id) = a$, where $id$ is the identity function on $\sigma(a)$
  • Consequence every normal element in a C*-algebra can be represented as a continuous function on its spectrum
• Spectral Theorem for Commutative C*-algebras every commutative C*-algebra $A$ is *-isomorphic to $C(X)$ for some compact Hausdorff space $X$
  • $X$ is homeomorphic to the space of characters (non-zero *-homomorphisms) on $A$
• Gelfand-Naimark Theorem every C*-algebra $A$ can be isometrically *-isomorphically embedded into $B(H)$ for some Hilbert space $H$
  • Consequence C*-algebras can be concretely represented as norm-closed *-subalgebras of bounded operators on a Hilbert space
• Functional Calculus Theorem for any normal element $a$ in a C*-algebra $A$ and any continuous function $f$ on $\sigma(a)$, there exists a unique element $f(a)$ in $C^*(a)$ such that $\Phi(f) = f(a)$, where $\Phi$ is the *-homomorphism from the Spectral Theorem
• Spectral Permanence Theorem if $\phi: A \to B$ is a -homomorphism between C-algebras and $a \in A$, then $\sigma(\phi(a)) \cup \{0\} = \phi(\sigma(a)) \cup \{0\}$
  • Consequence *-homomorphisms preserve spectra up to the possible addition of 0

Spectral Properties of Operators

• Spectrum of a self-adjoint operator is real, i.e., if $a = a^*$, then $\sigma(a) \subseteq \mathbb{R}$
• Spectrum of a unitary operator lies on the unit circle, i.e., if $u^*u = uu^* = 1$, then $\sigma(u) \subseteq \{z \in \mathbb{C} : |z| = 1\}$
• Spectrum of a positive operator is non-negative, i.e., if $a \geq 0$, then $\sigma(a) \subseteq [0, \infty)$
  • An element $a$ is positive if $a = b^*b$ for some $b$ in the C*-algebra
• Spectral radius formula $r(a) = \lim_{n \to \infty} \|a^n\|^{1/n}$ for any element $a$ in a C*-algebra
• Spectrum of a normal operator is contained in the numerical range, i.e., if $a$ is normal, then $\sigma(a) \subseteq \{(ax, x) : x \in H, \|x\| = 1\}$, where $H$ is the Hilbert space on which $a$ acts
• Compact operators have discrete spectrum, i.e., if $a$ is compact, then $\sigma(a) \setminus \{0\}$ consists only of eigenvalues with finite multiplicities
  • Multiplicities of eigenvalues are the dimensions of the corresponding eigenspaces

Continuous Functional Calculus

• Continuous Functional Calculus for any normal element $a$ in a C*-algebra $A$, there is a *-homomorphism $\Phi: C(\sigma(a)) \to C^*(a)$ such that $\Phi(id) = a$, where $id$ is the identity function on $\sigma(a)$
  • Allows for the definition of $f(a)$ for any continuous function $f$ on $\sigma(a)$
• Properties of the functional calculus
  • $\Phi$ is a unital *-homomorphism, i.e., $\Phi(1) = 1$ and $\Phi(f^*) = \Phi(f)^*$
  • $\Phi$ is norm-decreasing, i.e., $\|\Phi(f)\| \leq \|f\|_{\infty}$, where $\|f\|_{\infty}$ is the supremum norm of $f$ on $\sigma(a)$
  • $\Phi$ is spectrum-preserving, i.e., $\sigma(\Phi(f)) = f(\sigma(a))$ for any continuous function $f$ on $\sigma(a)$
• Holomorphic functional calculus an extension of the continuous functional calculus to holomorphic functions on a neighborhood of $\sigma(a)$
  • Allows for the definition of $f(a)$ for holomorphic functions $f$, not just continuous functions
• Borel functional calculus a further extension to bounded Borel functions on $\sigma(a)$
  • Requires the use of spectral measures and integration theory

Spectral Mapping Theorem

• Spectral Mapping Theorem for any normal element $a$ in a C*-algebra $A$ and any continuous function $f$ on $\sigma(a)$, we have $\sigma(f(a)) = f(\sigma(a))$
  • Consequence the spectrum of $f(a)$ is the image of the spectrum of $a$ under the function $f$
• Examples
  • If $a$ is self-adjoint and $f(t) = t^2$, then $\sigma(a^2) = \{\lambda^2 : \lambda \in \sigma(a)\}$
  • If $u$ is unitary and $f(z) = z^n$ for some integer $n$, then $\sigma(u^n) = \{z^n : z \in \sigma(u)\}$
• Spectral Mapping Theorem for holomorphic functions if $f$ is holomorphic on a neighborhood of $\sigma(a)$, then $\sigma(f(a)) = f(\sigma(a))$, where $f(a)$ is defined using the holomorphic functional calculus
• Spectral Mapping Theorem for Borel functions if $f$ is a bounded Borel function on $\sigma(a)$, then $\sigma(f(a)) = f(\sigma(a))$, where $f(a)$ is defined using the Borel functional calculus
  • Requires more advanced techniques from spectral theory and measure theory

Applications to C*-algebras

• Characterization of commutative C*-algebras a C*-algebra $A$ is commutative if and only if it is *-isomorphic to $C(X)$ for some compact Hausdorff space $X$
  • $X$ is uniquely determined up to homeomorphism by $A$
• Characterization of -homomorphisms between commutative C-algebras if $A$ and $B$ are commutative C*-algebras, then *-homomorphisms from $A$ to $B$ correspond bijectively to continuous functions from the character space of $B$ to the character space of $A$
• Functional calculus in C*-algebras the continuous functional calculus allows for the definition of $f(a)$ for any normal element $a$ in a C*-algebra $A$ and any continuous function $f$ on $\sigma(a)$
  • Useful for constructing new elements in C*-algebras with desired properties
• Spectral permanence in C*-algebras -homomorphisms between C-algebras preserve spectra, i.e., if $\phi: A \to B$ is a *-homomorphism and $a \in A$, then $\sigma(\phi(a)) \cup \{0\} = \phi(\sigma(a)) \cup \{0\}$
  • Useful for studying the behavior of spectra under mappings between C*-algebras
• Ideal structure of C*-algebras closed two-sided ideals in a C*-algebra $A$ correspond bijectively to open subsets of the primitive ideal space of $A$
  • Primitive ideal space consists of kernels of irreducible *-representations of $A$

Examples and Counterexamples

• Example commutative C*-algebras $C(X)$ for compact Hausdorff spaces $X$
  • Spectrum of $f \in C(X)$ is the range of $f$, i.e., $\sigma(f) = f(X)$
• Example non-commutative C*-algebras $B(H)$ for Hilbert spaces $H$, $M_n(\mathbb{C})$ for $n \geq 2$
  • Spectrum of an operator $T \in B(H)$ is not always equal to the range of $T$
• Example normal operators self-adjoint operators, unitary operators, projection operators
  • Spectral theorem applies to these operators
• Example non-normal operators shift operator on $\ell^2(\mathbb{N})$, nilpotent operators
  • Spectral theorem does not apply to these operators
• Counterexample spectrum is not always real even for self-adjoint elements in non-commutative C*-algebras
  • Consider a non-normal element $a$ with $a = a^*$ in a non-commutative C*-algebra
• Counterexample spectrum is not always non-negative even for positive elements in non-commutative C*-algebras
  • Consider a non-normal positive element $a$ in a non-commutative C*-algebra

Common Pitfalls and Misconceptions

• Misconception the spectrum of an element in a C*-algebra is always a subset of $\mathbb{R}$ or $\mathbb{C}$
  • The spectrum can be empty or unbounded in general
• Pitfall assuming the spectral theorem holds for all elements in a C*-algebra
  • The spectral theorem only applies to normal elements
• Misconception the spectrum of a self-adjoint element is always real
  • This is true in commutative C*-algebras but not necessarily in non-commutative C*-algebras
• Pitfall confusing the spectrum of an element with its numerical range
  • The numerical range contains the spectrum but can be strictly larger
• Misconception the functional calculus always produces a normal element
  • The functional calculus preserves normality, but $f(a)$ may not be normal if $f$ is not continuous or if $a$ is not normal
• Pitfall assuming *-homomorphisms always preserve spectra exactly
  • *-homomorphisms preserve spectra up to the possible addition of 0
• Misconception the spectral radius is always equal to the operator norm
  • The spectral radius is always less than or equal to the operator norm, with equality for normal elements
• Pitfall forgetting the importance of the C*-identity $\|a^*a\| = \|a\|^2$
  • Many results in spectral theory rely crucially on this identity, which distinguishes C*-algebras from general Banach algebras