Spectral theory of normal C*-algebras

In functional analysis, every C^*-algebra is isomorphic to a subalgebra of the C^*-algebra [math]\displaystyle{ \mathcal{B}(H) }[/math] of bounded linear operators on some Hilbert space [math]\displaystyle{ H. }[/math] This article describes the spectral theory of closed normal subalgebras of [math]\displaystyle{ \mathcal{B}(H) }[/math]. A subalgebra [math]\displaystyle{ A }[/math] of [math]\displaystyle{ \mathcal{B}(H) }[/math] is called normal if it is commutative and closed under the [math]\displaystyle{ \ast }[/math] operation: for all [math]\displaystyle{ x,y\in A }[/math], we have [math]\displaystyle{ x^\ast\in A }[/math] and that [math]\displaystyle{ xy = yx }[/math].

Resolution of identity

Throughout, [math]\displaystyle{ H }[/math] is a fixed Hilbert space.

A projection-valued measure on a measurable space [math]\displaystyle{ (X, \Omega), }[/math] where [math]\displaystyle{ \Omega }[/math] is a σ-algebra of subsets of [math]\displaystyle{ X, }[/math] is a mapping [math]\displaystyle{ \pi : \Omega \to \mathcal{B}(H) }[/math] such that for all [math]\displaystyle{ \omega \in \Omega, }[/math] [math]\displaystyle{ \pi(\omega) }[/math] is a self-adjoint projection on [math]\displaystyle{ H }[/math] (that is, [math]\displaystyle{ \pi(\omega) }[/math] is a bounded linear operator [math]\displaystyle{ \pi(\omega) : H \to H }[/math] that satisfies [math]\displaystyle{ \pi(\omega) = \pi(\omega)^* }[/math] and [math]\displaystyle{ \pi(\omega) \circ \pi(\omega) = \pi(\omega) }[/math]) such that [math]\displaystyle{ \pi(X) = \operatorname{Id}_H \quad }[/math] (where [math]\displaystyle{ \operatorname{Id}_H }[/math] is the identity operator of [math]\displaystyle{ H }[/math]) and for every [math]\displaystyle{ x, y \in H, }[/math] the function [math]\displaystyle{ \Omega \to \Complex }[/math] defined by [math]\displaystyle{ \omega \mapsto \langle \pi(\omega)x, y \rangle }[/math] is a complex measure on [math]\displaystyle{ M }[/math] (that is, a complex-valued countably additive function).

A resolution of identity on a measurable space [math]\displaystyle{ (X, \Omega) }[/math] is a function [math]\displaystyle{ \pi : \Omega \to \mathcal{B}(H) }[/math] such that for every [math]\displaystyle{ \omega_1, \omega_2 \in \Omega }[/math]:

1. [math]\displaystyle{ \pi(\varnothing) = 0 }[/math];
2. [math]\displaystyle{ \pi(X) = \operatorname{Id}_H }[/math];
3. for every [math]\displaystyle{ \omega \in \Omega, }[/math] [math]\displaystyle{ \pi(\omega) }[/math] is a self-adjoint projection on [math]\displaystyle{ H }[/math];
4. for every [math]\displaystyle{ x, y \in H, }[/math] the map [math]\displaystyle{ \pi_{x, y} : \Omega \to \Complex }[/math] defined by [math]\displaystyle{ \pi_{x,y}(\omega) = \langle \pi(\omega) x, y \rangle }[/math] is a complex measure on [math]\displaystyle{ \Omega }[/math];
5. [math]\displaystyle{ \pi\left(\omega_1 \cap \omega_2\right) = \pi\left(\omega_1\right) \circ \pi\left(\omega_2\right) }[/math];
6. if [math]\displaystyle{ \omega_1 \cap \omega_2 = \varnothing }[/math] then [math]\displaystyle{ \pi \left(\omega_1 \cup \omega_2\right) = \pi\left(\omega_1\right) + \pi\left(\omega_2\right) }[/math];

If [math]\displaystyle{ \Omega }[/math] is the [math]\displaystyle{ \sigma }[/math]-algebra of all Borels sets on a Hausdorff locally compact (or compact) space, then the following additional requirement is added:

7. for every [math]\displaystyle{ x, y \in H, }[/math] the map [math]\displaystyle{ \pi_{x, y} : \Omega \to \Complex }[/math] is a regular Borel measure (this is automatically satisfied on compact metric spaces).

Conditions 2, 3, and 4 imply that [math]\displaystyle{ \pi }[/math] is a projection-valued measure.

Properties

Throughout, let [math]\displaystyle{ \pi }[/math] be a resolution of identity. For all [math]\displaystyle{ x \in H, }[/math] [math]\displaystyle{ \pi_{x, x} : \Omega \to \Complex }[/math] is a positive measure on [math]\displaystyle{ \Omega }[/math] with total variation [math]\displaystyle{ \left\|\pi_{x, x}\right\| = \pi_{x, x}(X) = \|x\|^2 }[/math] and that satisfies [math]\displaystyle{ \pi_{x, x}(\omega) = \langle \pi(\omega) x, x \rangle = \|\pi(\omega) x\|^2 }[/math] for all [math]\displaystyle{ \omega \in \Omega. }[/math]

For every [math]\displaystyle{ \omega_1, \omega_2 \in \Omega }[/math]:

• [math]\displaystyle{ \pi\left(\omega_1\right) \pi\left(\omega_2\right) = \pi\left(\omega_2\right) \pi\left(\omega_1\right) }[/math] (since both are equal to [math]\displaystyle{ \pi\left(\omega_1 \cap \omega_2\right) }[/math]).
• If [math]\displaystyle{ \omega_1 \cap \omega_2 = \varnothing }[/math] then the ranges of the maps [math]\displaystyle{ \pi\left(\omega_1\right) }[/math] and [math]\displaystyle{ \pi\left(\omega_2\right) }[/math] are orthogonal to each other and [math]\displaystyle{ \pi\left(\omega_1\right) \pi\left(\omega_2\right) = 0 = \pi\left(\omega_2\right) \pi\left(\omega_1\right). }[/math]
• [math]\displaystyle{ \pi : \Omega \to \mathcal{B}(H) }[/math] is finitely additive.
• If [math]\displaystyle{ \omega_1, \omega_2, \ldots }[/math] are pairwise disjoint elements of [math]\displaystyle{ \Omega }[/math] whose union is [math]\displaystyle{ \omega }[/math] and if [math]\displaystyle{ \pi\left(\omega_i\right) = 0 }[/math] for all [math]\displaystyle{ i }[/math] then [math]\displaystyle{ \pi(\omega) = 0. }[/math]
• However, [math]\displaystyle{ \pi : \Omega \to \mathcal{B}(H) }[/math] is countably additive only in trivial situations as is now described: suppose that [math]\displaystyle{ \omega_1, \omega_2, \ldots }[/math] are pairwise disjoint elements of [math]\displaystyle{ \Omega }[/math] whose union is [math]\displaystyle{ \omega }[/math] and that the partial sums [math]\displaystyle{ \sum_{i=1}^n \pi\left(\omega_i\right) }[/math] converge to [math]\displaystyle{ \pi(\omega) }[/math] in [math]\displaystyle{ \mathcal{B}(H) }[/math] (with its norm topology) as [math]\displaystyle{ n \to \infty }[/math]; then since the norm of any projection is either [math]\displaystyle{ 0 }[/math] or [math]\displaystyle{ \geq 1, }[/math] the partial sums cannot form a Cauchy sequence unless all but finitely many of the [math]\displaystyle{ \pi\left(\omega_i\right) }[/math] are [math]\displaystyle{ 0. }[/math]
• For any fixed [math]\displaystyle{ x \in H, }[/math] the map [math]\displaystyle{ \pi_x : \Omega \to H }[/math] defined by [math]\displaystyle{ \pi_{x}(\omega) := \pi(\omega) x }[/math] is a countably additive [math]\displaystyle{ H }[/math]-valued measure on [math]\displaystyle{ \Omega. }[/math]
  • Here countably additive means that whenever [math]\displaystyle{ \omega_1, \omega_2, \ldots }[/math] are pairwise disjoint elements of [math]\displaystyle{ \Omega }[/math] whose union is [math]\displaystyle{ \omega, }[/math] then the partial sums [math]\displaystyle{ \sum_{i=1}^n \pi\left(\omega_i\right) x }[/math] converge to [math]\displaystyle{ \pi(\omega)x }[/math] in [math]\displaystyle{ H. }[/math] Said more succinctly, [math]\displaystyle{ \sum_{i=1}^{\infty} \pi\left(\omega_i\right) x = \pi(\omega) x. }[/math]
  • In other words, for every pairwise disjoint family of elements [math]\displaystyle{ \left(\omega_i\right)_{i = 1}^{\infty}\subseteq \Omega }[/math] whose union is [math]\displaystyle{ \omega_{\infty}\in\Omega }[/math], then [math]\displaystyle{ \sum_{i = 1}^{n} \pi\left(\omega_i\right) = \pi\left(\bigcup_{i=1}^{n} \omega_i\right) }[/math] (by finite additivity of [math]\displaystyle{ \pi }[/math]) converges to [math]\displaystyle{ \pi\left(\omega_{\infty}\right) }[/math] in the strong operator topology on [math]\displaystyle{ \mathcal{B}(H) }[/math]: for every [math]\displaystyle{ x\in H }[/math], the sequence of elements [math]\displaystyle{ \sum_{i=1}^{n}\pi\left(\omega_i\right)x }[/math] converges to [math]\displaystyle{ \pi\left(\omega_{\infty}\right)x }[/math] in [math]\displaystyle{ H }[/math] (with respect to the norm topology).

L^∞(π) - space of essentially bounded function

The [math]\displaystyle{ \pi : \Omega \to \mathcal{B}(H) }[/math] be a resolution of identity on [math]\displaystyle{ (X, \Omega). }[/math]

Essentially bounded functions

Suppose [math]\displaystyle{ f : X \to \Complex }[/math] is a complex-valued [math]\displaystyle{ \Omega }[/math]-measurable function. There exists a unique largest open subset [math]\displaystyle{ V_f }[/math] of [math]\displaystyle{ \Complex }[/math] (ordered under subset inclusion) such that [math]\displaystyle{ \pi\left(f^{-1}\left(V_f\right)\right) = 0. }[/math] To see why, let [math]\displaystyle{ D_1, D_2, \ldots }[/math] be a basis for [math]\displaystyle{ \Complex }[/math]'s topology consisting of open disks and suppose that [math]\displaystyle{ D_{i_1}, D_{i_2}, \ldots }[/math] is the subsequence (possibly finite) consisting of those sets such that [math]\displaystyle{ \pi\left(f^{-1}\left(D_{i_k}\right)\right) = 0 }[/math]; then [math]\displaystyle{ D_{i_1} \cup D_{i_2} \cup \cdots = V_f. }[/math] Note that, in particular, if [math]\displaystyle{ D }[/math] is an open subset of [math]\displaystyle{ \Complex }[/math] such that [math]\displaystyle{ D \cap \operatorname{Im} f = \varnothing }[/math] then [math]\displaystyle{ \pi\left(f^{-1}(D)\right) = \pi (\varnothing) = 0 }[/math] so that [math]\displaystyle{ D \subseteq V_f }[/math] (although there are other ways in which [math]\displaystyle{ \pi\left(f^{-1}(D)\right) }[/math] may equal 0). Indeed, [math]\displaystyle{ \Complex \setminus \operatorname{cl}(\operatorname{Im} f) \subseteq V_f. }[/math]

The essential range of [math]\displaystyle{ f }[/math] is defined to be the complement of [math]\displaystyle{ V_f. }[/math] It is the smallest closed subset of [math]\displaystyle{ \Complex }[/math] that contains [math]\displaystyle{ f(x) }[/math] for almost all [math]\displaystyle{ x \in X }[/math] (that is, for all [math]\displaystyle{ x \in X }[/math] except for those in some set [math]\displaystyle{ \omega \in \Omega }[/math] such that [math]\displaystyle{ \pi(\omega) = 0 }[/math]). The essential range is a closed subset of [math]\displaystyle{ \Complex }[/math] so that if it is also a bounded subset of [math]\displaystyle{ \Complex }[/math] then it is compact.

The function [math]\displaystyle{ f }[/math] is essentially bounded if its essential range is bounded, in which case define its essential supremum, denoted by [math]\displaystyle{ \|f\|^{\infty}, }[/math] to be the supremum of all [math]\displaystyle{ |\lambda| }[/math] as [math]\displaystyle{ \lambda }[/math] ranges over the essential range of [math]\displaystyle{ f. }[/math]

Space of essentially bounded functions

Let [math]\displaystyle{ \mathcal{B}(X, \Omega) }[/math] be the vector space of all bounded complex-valued [math]\displaystyle{ \Omega }[/math]-measurable functions [math]\displaystyle{ f : X \to \Complex, }[/math] which becomes a Banach algebra when normed by [math]\displaystyle{ \|f\|_{\infty} := \sup_{x \in X}|f(x) |. }[/math] The function [math]\displaystyle{ \|\,\cdot\,\|^{\infty} }[/math] is a seminorm on [math]\displaystyle{ \mathcal{B}(X, \Omega), }[/math] but not necessarily a norm. The kernel of this seminorm, [math]\displaystyle{ N^{\infty} := \left\{ f \in \mathcal{B}(X, \Omega) : \|f\|^{\infty} = 0 \right\}, }[/math] is a vector subspace of [math]\displaystyle{ \mathcal{B}(X, \Omega) }[/math] that is a closed two-sided ideal of the Banach algebra [math]\displaystyle{ \left(\mathcal{B}(X, \Omega), \| \cdot \|_{\infty}\right). }[/math] Hence the quotient of [math]\displaystyle{ \mathcal{B}(X, \Omega) }[/math] by [math]\displaystyle{ N^{\infty} }[/math] is also a Banach algebra, denoted by [math]\displaystyle{ L^{\infty}(\pi) := \mathcal{B}(X, \Omega) / N^{\infty} }[/math] where the norm of any element [math]\displaystyle{ f + N^{\infty} \in L^{\infty}(\pi) }[/math] is equal to [math]\displaystyle{ \|f\|^{\infty} }[/math] (since if [math]\displaystyle{ f + N^{\infty} = g + N^{\infty} }[/math] then [math]\displaystyle{ \|f\|^{\infty} = \| g \|^{\infty} }[/math]) and this norm makes [math]\displaystyle{ L^{\infty}(\pi) }[/math] into a Banach algebra. The spectrum of [math]\displaystyle{ f + N^{\infty} }[/math] in [math]\displaystyle{ L^{\infty}(\pi) }[/math] is the essential range of [math]\displaystyle{ f. }[/math] This article will follow the usual practice of writing [math]\displaystyle{ f }[/math] rather than [math]\displaystyle{ f + N^{\infty} }[/math] to represent elements of [math]\displaystyle{ L^{\infty}(\pi). }[/math]

Theorem — Let [math]\displaystyle{ \pi : \Omega \to \mathcal{B}(H) }[/math] be a resolution of identity on [math]\displaystyle{ (X, \Omega). }[/math] There exists a closed normal subalgebra [math]\displaystyle{ A }[/math] of [math]\displaystyle{ \mathcal{B}(H) }[/math] and an isometric ^*-isomorphism [math]\displaystyle{ \Psi : L^{\infty}(\pi) \to A }[/math] satisfying the following properties:

1. [math]\displaystyle{ \langle \Psi(f) x, y \rangle = \int_X f \operatorname{d} \pi_{x, y} }[/math] for all [math]\displaystyle{ x, y \in H }[/math] and [math]\displaystyle{ f \in L^{\infty}(\pi), }[/math] which justifies the notation [math]\displaystyle{ \Psi(f) = \int_X f \operatorname{d}\pi }[/math];
2. [math]\displaystyle{ \|\Psi(f) x\|^2 = \int_X|f|^2 \operatorname{d} \pi_{x, x} }[/math] for all [math]\displaystyle{ x \in H }[/math] and [math]\displaystyle{ f \in L^{\infty}(\pi) }[/math];
3. an operator [math]\displaystyle{ R \in \mathbb{B}(H) }[/math] commutes with every element of [math]\displaystyle{ \operatorname{Im} \pi }[/math] if and only if it commutes with every element of [math]\displaystyle{ A = \operatorname{Im} \Psi. }[/math]
4. if [math]\displaystyle{ f }[/math] is a simple function equal to [math]\displaystyle{ f = \sum_{i=1}^n \lambda_i \mathbb{1}_{\omega_i}, }[/math] where [math]\displaystyle{ \omega_1, \ldots \omega_n }[/math] is a partition of [math]\displaystyle{ X }[/math] and the [math]\displaystyle{ \lambda_i }[/math] are complex numbers, then [math]\displaystyle{ \Psi(f) = \sum_{i=1}^n \lambda_i \pi\left(\omega_i\right) }[/math] (here [math]\displaystyle{ \mathbb{1} }[/math] is the characteristic function);
5. if [math]\displaystyle{ f }[/math] is the limit (in the norm of [math]\displaystyle{ L^{\infty}(\pi) }[/math]) of a sequence of simple functions [math]\displaystyle{ s_1, s_2, \ldots }[/math] in [math]\displaystyle{ L^{\infty}(\pi) }[/math] then [math]\displaystyle{ \left(\Psi\left(s_i\right)\right)_{i=1}^{\infty} }[/math] converges to[math]\displaystyle{ \Psi(f) }[/math] in [math]\displaystyle{ \mathcal{B}(H) }[/math] and [math]\displaystyle{ \|\Psi(f)\| = \|f\|^{\infty} }[/math];
6. [math]\displaystyle{ \left(\|f\|^{\infty}\right)^2 = \sup_{\|h\| \leq 1} \int_X \operatorname{d} \pi_{h, h} }[/math] for every [math]\displaystyle{ f \in L^{\infty}(\pi). }[/math]

Spectral theorem

The maximal ideal space of a Banach algebra [math]\displaystyle{ A }[/math] is the set of all complex homomorphisms [math]\displaystyle{ A \to \Complex, }[/math] which we'll denote by [math]\displaystyle{ \sigma_A. }[/math] For every [math]\displaystyle{ T }[/math] in [math]\displaystyle{ A, }[/math] the Gelfand transform of [math]\displaystyle{ T }[/math] is the map [math]\displaystyle{ G(T) : \sigma_A \to \Complex }[/math] defined by [math]\displaystyle{ G(T)(h) := h(T). }[/math] [math]\displaystyle{ \sigma_A }[/math] is given the weakest topology making every [math]\displaystyle{ G(T) : \sigma_A \to \Complex }[/math] continuous. With this topology, [math]\displaystyle{ \sigma_A }[/math] is a compact Hausdorff space and every [math]\displaystyle{ T }[/math] in [math]\displaystyle{ A, }[/math] [math]\displaystyle{ G(T) }[/math] belongs to [math]\displaystyle{ C \left(\sigma_A\right), }[/math] which is the space of continuous complex-valued functions on [math]\displaystyle{ \sigma_A. }[/math] The range of [math]\displaystyle{ G(T) }[/math] is the spectrum [math]\displaystyle{ \sigma(T) }[/math] and that the spectral radius is equal to [math]\displaystyle{ \max \left\{ |G(T)(h)|: h \in \sigma_A \right\}, }[/math] which is [math]\displaystyle{ \leq \|T\|. }[/math]

Theorem — Suppose [math]\displaystyle{ A }[/math] is a closed normal subalgebra of [math]\displaystyle{ \mathcal{B}(H) }[/math] that contains the identity operator [math]\displaystyle{ \operatorname{Id}_H }[/math] and let [math]\displaystyle{ \sigma = \sigma_A }[/math] be the maximal ideal space of [math]\displaystyle{ A. }[/math] Let [math]\displaystyle{ \Omega }[/math] be the Borel subsets of [math]\displaystyle{ \sigma. }[/math] For every [math]\displaystyle{ T }[/math] in [math]\displaystyle{ A, }[/math] let [math]\displaystyle{ G(T) : \sigma_A \to \Complex }[/math] denote the Gelfand transform of [math]\displaystyle{ T }[/math] so that [math]\displaystyle{ G }[/math] is an injective map [math]\displaystyle{ G : A \to C\left(\sigma_A\right). }[/math] There exists a unique resolution of identity [math]\displaystyle{ \pi : \Omega \to A }[/math] that satisfies: [math]\displaystyle{ \langle T x, y \rangle = \int_{\sigma_A} G(T) \operatorname{d} \pi_{x, y} \quad \text{ for all } x, y \in H \text{ and all } T \in A; }[/math] the notation [math]\displaystyle{ T = \int_{\sigma_A} G(T) \operatorname{d} \pi }[/math] is used to summarize this situation. Let [math]\displaystyle{ I : \operatorname{Im} G \to A }[/math] be the inverse of the Gelfand transform [math]\displaystyle{ G : A \to C\left(\sigma_A\right) }[/math] where [math]\displaystyle{ \operatorname{Im} G }[/math] can be canonically identified as a subspace of [math]\displaystyle{ L^{\infty}(\pi). }[/math] Let [math]\displaystyle{ B }[/math] be the closure (in the norm topology of [math]\displaystyle{ \mathcal{B}(H) }[/math]) of the linear span of [math]\displaystyle{ \operatorname{Im} \pi. }[/math] Then the following are true:

1. [math]\displaystyle{ B }[/math] is a closed subalgebra of [math]\displaystyle{ \mathcal{B}(H) }[/math] containing [math]\displaystyle{ A. }[/math]
2. There exists a (linear multiplicative) isometric ^*-isomorphism [math]\displaystyle{ \Phi : L^{\infty}(\pi) \to B }[/math] extending [math]\displaystyle{ I : \operatorname{Im} G \to A }[/math] such that [math]\displaystyle{ \Phi f = \int_{\sigma_A} f \operatorname{d} \pi }[/math] for all [math]\displaystyle{ f \in L^{\infty}(\pi). }[/math]
   • Recall that the notation [math]\displaystyle{ \Phi f = \int_{\sigma_A} f \operatorname{d} \pi }[/math] means that [math]\displaystyle{ \langle (\Phi f) x, y \rangle = \int_{\sigma_A} f \operatorname{d} \pi_{x, y} }[/math] for all [math]\displaystyle{ x, y \in H }[/math];
   • Note in particular that [math]\displaystyle{ T = \int_{\sigma_A} G(T) \operatorname{d} \pi = \Phi(G(T)) }[/math] for all [math]\displaystyle{ T \in A. }[/math]
   • Explicitly, [math]\displaystyle{ \Phi }[/math] satisfies [math]\displaystyle{ \Phi \left(\overline{f}\right) = (\Phi f)^* }[/math] and [math]\displaystyle{ \|\Phi f\| = \|f\|^{\infty} }[/math] for every [math]\displaystyle{ f \in L^{\infty}(\pi) }[/math] (so if [math]\displaystyle{ f }[/math] is real valued then [math]\displaystyle{ \Phi(f) }[/math] is self-adjoint).
3. If [math]\displaystyle{ \omega \subseteq \sigma_A }[/math] is open and nonempty (which implies that [math]\displaystyle{ \omega \in \Omega }[/math]) then [math]\displaystyle{ \pi(\omega) \neq 0. }[/math]
4. A bounded linear operator [math]\displaystyle{ S \in \mathcal{B}(H) }[/math] commutes with every element of [math]\displaystyle{ A }[/math] if and only if it commutes with every element of [math]\displaystyle{ \operatorname{Im} \pi. }[/math]

The above result can be specialized to a single normal bounded operator.

See also

• Projection-valued measure – Mathematical operator-value measure of interest in quantum mechanics and functional analysis
• Spectral theory of compact operators
• Spectral theorem – Result about when a matrix can be diagonalized

References

• Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250. 
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250. 
• Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.