Spectral theory of normal C*-algebras

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

Resolution of identity

Throughout, $H$ is a fixed Hilbert space.

A projection-valued measure on a measurable space $(X,\mathrm{\Omega }),$ where $\mathrm{\Omega }$ is a σ-algebra of subsets of $X,$ is a mapping $\pi :\mathrm{\Omega }\to {ℬ}(H)$ such that for all $\omega \in \mathrm{\Omega },$ $\pi (\omega )$ is a self-adjoint projection on $H$ (that is, $\pi (\omega )$ is a bounded linear operator $\pi (\omega ):H\to H$ that satisfies $\pi (\omega )=\pi (\omega {)}^{*}$ and $\pi (\omega )\circ \pi (\omega )=\pi (\omega )$) such that $$\pi (X)={\mathrm{Id}}_H$$ (where ${\mathrm{Id}}_H$ is the identity operator of $H$) and for every $x,y\in H,$ the function $\mathrm{\Omega }\to \mathbb{ℂ}$ defined by $\omega \mapsto ⟨\pi (\omega )x,y⟩$ is a complex measure on $M$ (that is, a complex-valued countably additive function).

A resolution of identity on a measurable space $(X,\mathrm{\Omega })$ is a function $\pi :\mathrm{\Omega }\to {ℬ}(H)$ such that for every ${\omega }_1,{\omega }_2\in \mathrm{\Omega }$:

1. $\pi (\varnothing )=0$;
2. $\pi (X)={\mathrm{Id}}_H$;
3. for every $\omega \in \mathrm{\Omega },$ $\pi (\omega )$ is a self-adjoint projection on $H$;
4. for every $x,y\in H,$ the map ${\pi }_{x,y}:\mathrm{\Omega }\to \mathbb{ℂ}$ defined by ${\pi }_{x,y}(\omega )=⟨\pi (\omega )x,y⟩$ is a complex measure on $\mathrm{\Omega }$;
5. $\pi ({\omega }_1\cap {\omega }_2)=\pi \left({\omega }_1\right)\circ \pi \left({\omega }_2\right)$;
6. if ${\omega }_1\cap {\omega }_2=\varnothing$ then $\pi ({\omega }_1\cup {\omega }_2)=\pi \left({\omega }_1\right)+\pi \left({\omega }_2\right)$;

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

7. for every $x,y\in H,$ the map ${\pi }_{x,y}:\mathrm{\Omega }\to \mathbb{ℂ}$ is a regular Borel measure (this is automatically satisfied on compact metric spaces).

Conditions 2, 3, and 4 imply that $\pi$ is a projection-valued measure.

Properties

Throughout, let $\pi$ be a resolution of identity. For all $x\in H,$ ${\pi }_{x,x}:\mathrm{\Omega }\to \mathbb{ℂ}$ is a positive measure on $\mathrm{\Omega }$ with total variation $\Vert {\pi }_{x,x}\Vert ={\pi }_{x,x}(X)=\Vert x{\Vert }^2$ and that satisfies ${\pi }_{x,x}(\omega )=⟨\pi (\omega )x,x⟩=\Vert \pi (\omega )x{\Vert }^2$ for all $\omega \in \mathrm{\Omega }.$

For every ${\omega }_1,{\omega }_2\in \mathrm{\Omega }$:

• $\pi \left({\omega }_1\right)\pi \left({\omega }_2\right)=\pi \left({\omega }_2\right)\pi \left({\omega }_1\right)$ (since both are equal to $\pi ({\omega }_1\cap {\omega }_2)$).
• If ${\omega }_1\cap {\omega }_2=\varnothing$ then the ranges of the maps $\pi \left({\omega }_1\right)$ and $\pi \left({\omega }_2\right)$ are orthogonal to each other and $\pi \left({\omega }_1\right)\pi \left({\omega }_2\right)=0=\pi \left({\omega }_2\right)\pi \left({\omega }_1\right).$
• $\pi :\mathrm{\Omega }\to {ℬ}(H)$ is finitely additive.
• If ${\omega }_1,{\omega }_2,\dots$ are pairwise disjoint elements of $\mathrm{\Omega }$ whose union is $\omega$ and if $\pi \left({\omega }_i\right)=0$ for all $i$ then $\pi (\omega )=0.$
• However, $\pi :\mathrm{\Omega }\to {ℬ}(H)$ is countably additive only in trivial situations as is now described: suppose that ${\omega }_1,{\omega }_2,\dots$ are pairwise disjoint elements of $\mathrm{\Omega }$ whose union is $\omega$ and that the partial sums $\sum _{i=1}^n\pi \left({\omega }_i\right)$ converge to $\pi (\omega )$ in ${ℬ}(H)$ (with its norm topology) as $n\to \mathrm{\infty }$; then since the norm of any projection is either $0$ or $\ge 1,$ the partial sums cannot form a Cauchy sequence unless all but finitely many of the $\pi \left({\omega }_i\right)$ are $0.$
• For any fixed $x\in H,$ the map ${\pi }_x:\mathrm{\Omega }\to H$ defined by ${\pi }_x(\omega ):=\pi (\omega )x$ is a countably additive $H$-valued measure on $\mathrm{\Omega }.$
  • Here countably additive means that whenever ${\omega }_1,{\omega }_2,\dots$ are pairwise disjoint elements of $\mathrm{\Omega }$ whose union is $\omega ,$ then the partial sums $\sum _{i=1}^n\pi \left({\omega }_i\right)x$ converge to $\pi (\omega )x$ in $H.$ Said more succinctly, $\sum _{i=1}^{\mathrm{\infty }}\pi \left({\omega }_i\right)x=\pi (\omega )x.$
  • In other words, for every pairwise disjoint family of elements ${\left({\omega }_i\right)}_{i=1}^{\mathrm{\infty }}\subseteq \mathrm{\Omega }$ whose union is ${\omega }_{\mathrm{\infty }}\in \mathrm{\Omega }$, then $\sum _{i=1}^n\pi \left({\omega }_i\right)=\pi \left(\bigcup _{i=1}^n{\omega }_i\right)$ (by finite additivity of $\pi$) converges to $\pi \left({\omega }_{\mathrm{\infty }}\right)$ in the strong operator topology on ${ℬ}(H)$: for every $x\in H$, the sequence of elements $\sum _{i=1}^n\pi \left({\omega }_i\right)x$ converges to $\pi \left({\omega }_{\mathrm{\infty }}\right)x$ in $H$ (with respect to the norm topology).

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

The $\pi :\mathrm{\Omega }\to {ℬ}(H)$ be a resolution of identity on $(X,\mathrm{\Omega }).$

Essentially bounded functions

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

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

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

Space of essentially bounded functions

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

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

1. $⟨\mathrm{\Psi }(f)x,y⟩=\underset{X}{\int }f\mathrm{d}{\pi }_{x,y}$ for all $x,y\in H$ and $f\in L^{\mathrm{\infty }}(\pi ),$ which justifies the notation $\mathrm{\Psi }(f)=\underset{X}{\int }f\mathrm{d}\pi$;
2. $\Vert \mathrm{\Psi }(f)x{\Vert }^2=\underset{X}{\int }|f{|}^2\mathrm{d}{\pi }_{x,x}$ for all $x\in H$ and $f\in L^{\mathrm{\infty }}(\pi )$;
3. an operator $R\in \mathbb{𝔹}(H)$ commutes with every element of $\mathrm{Im}\pi$ if and only if it commutes with every element of $A=\mathrm{Im}\mathrm{\Psi }.$
4. if $f$ is a simple function equal to $f=\sum _{i=1}^n{\lambda }_i{\mathbb{𝟙}}_{{\omega }_i},$ where ${\omega }_1,\dots {\omega }_n$ is a partition of $X$ and the ${\lambda }_i$ are complex numbers, then $\mathrm{\Psi }(f)=\sum _{i=1}^n{\lambda }_i\pi \left({\omega }_i\right)$ (here $\mathbb{𝟙}$ is the characteristic function);
5. if $f$ is the limit (in the norm of $L^{\mathrm{\infty }}(\pi )$) of a sequence of simple functions $s_1,s_2,\dots$ in $L^{\mathrm{\infty }}(\pi )$ then ${\left(\mathrm{\Psi }\left(s_i\right)\right)}_{i=1}^{\mathrm{\infty }}$ converges to$\mathrm{\Psi }(f)$ in ${ℬ}(H)$ and $\Vert \mathrm{\Psi }(f)\Vert =\Vert f{\Vert }^{\mathrm{\infty }}$;
6. ${(\Vert f{\Vert }^{\mathrm{\infty }})}^2=\underset{\Vert h\Vert \le 1}{\sup }\underset{X}{\int }\mathrm{d}{\pi }_{h,h}$ for every $f\in L^{\mathrm{\infty }}(\pi ).$

Spectral theorem

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

Theorem — Suppose $A$ is a closed normal subalgebra of ${ℬ}(H)$ that contains the identity operator ${\mathrm{Id}}_H$ and let $\sigma ={\sigma }_A$ be the maximal ideal space of $A.$ Let $\mathrm{\Omega }$ be the Borel subsets of $\sigma .$ For every $T$ in $A,$ let $G(T):{\sigma }_A\to \mathbb{ℂ}$ denote the Gelfand transform of $T$ so that $G$ is an injective map $G:A\to C\left({\sigma }_A\right).$ There exists a unique resolution of identity $\pi :\mathrm{\Omega }\to A$ that satisfies: $$⟨Tx,y⟩=\underset{{\sigma }_A}{\int }G(T)\mathrm{d}{\pi }_{x,y}\text{ for all }x,y\in H\text{ and all }T\in A;$$ the notation $T=\underset{{\sigma }_A}{\int }G(T)\mathrm{d}\pi$ is used to summarize this situation. Let $I:\mathrm{Im}G\to A$ be the inverse of the Gelfand transform $G:A\to C\left({\sigma }_A\right)$ where $\mathrm{Im}G$ can be canonically identified as a subspace of $L^{\mathrm{\infty }}(\pi ).$ Let $B$ be the closure (in the norm topology of ${ℬ}(H)$) of the linear span of $\mathrm{Im}\pi .$ Then the following are true:

1. $B$ is a closed subalgebra of ${ℬ}(H)$ containing $A.$
2. There exists a (linear multiplicative) isometric ^*-isomorphism $\mathrm{\Phi }:L^{\mathrm{\infty }}(\pi )\to B$ extending $I:\mathrm{Im}G\to A$ such that $\mathrm{\Phi }f=\underset{{\sigma }_A}{\int }f\mathrm{d}\pi$ for all $f\in L^{\mathrm{\infty }}(\pi ).$
   • Recall that the notation $\mathrm{\Phi }f=\underset{{\sigma }_A}{\int }f\mathrm{d}\pi$ means that $⟨(\mathrm{\Phi }f)x,y⟩=\underset{{\sigma }_A}{\int }f\mathrm{d}{\pi }_{x,y}$ for all $x,y\in H$;
   • Note in particular that $T=\underset{{\sigma }_A}{\int }G(T)\mathrm{d}\pi =\mathrm{\Phi }(G(T))$ for all $T\in A.$
   • Explicitly, $\mathrm{\Phi }$ satisfies $\mathrm{\Phi }\left(\bar{f}\right)=(\mathrm{\Phi }f{)}^{*}$ and $\Vert \mathrm{\Phi }f\Vert =\Vert f{\Vert }^{\mathrm{\infty }}$ for every $f\in L^{\mathrm{\infty }}(\pi )$ (so if $f$ is real valued then $\mathrm{\Phi }(f)$ is self-adjoint).
3. If $\omega \subseteq {\sigma }_A$ is open and nonempty (which implies that $\omega \in \mathrm{\Omega }$) then $\pi (\omega )\ne 0.$
4. A bounded linear operator $S\in {ℬ}(H)$ commutes with every element of $A$ if and only if it commutes with every element of $\mathrm{Im}\pi .$

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.