# Spectral theory of normal C*-algebras

In functional analysis, every C$^*$-algebra is isomorphic to a subalgebra of the C$^*$-algebra ${\mathcal {B}}(H)$ of bounded linear operators on some Hilbert space $H.$ This article describes the spectral theory of closed normal subalgebras of ${\mathcal {B}}(H)$. A subalgebra $A$ of ${\mathcal {B}}(H)$ 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 $(X,\Omega ),$ where $\Omega$ is a σ-algebra of subsets of $X,$ is a mapping $\pi :\Omega \to {\mathcal {B}}(H)$ such that for all $\omega \in \Omega ,$ $\pi (\omega )$ 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 $\pi (X)=\operatorname {Id} _{H}\quad$ (where $\operatorname {Id} _{H}$ is the identity operator of $H$) and for every $x,y\in H,$ the function $\Omega \to \mathbb {C}$ defined by $\omega \mapsto \langle \pi (\omega )x,y\rangle$ is a complex measure on $M$ (that is, a complex-valued countably additive function).
A resolution of identity on a measurable space $(X,\Omega )$ is a function $\pi :\Omega \to {\mathcal {B}}(H)$ such that for every $\omega _{1},\omega _{2}\in \Omega$:
* $\pi (\varnothing )=0$;
* $\pi (X)=\operatorname {Id} _{H}$;
* for every $\omega \in \Omega ,$ $\pi (\omega )$ is a self-adjoint projection on $H$;
* for every $x,y\in H,$ the map $\pi _{x,y}:\Omega \to \mathbb {C}$ defined by $\pi _{x,y}(\omega )=\langle \pi (\omega )x,y\rangle$ is a complex measure on $\Omega$;
* $\pi \left(\omega _{1}\cap \omega _{2}\right)=\pi \left(\omega _{1}\right)\circ \pi \left(\omega _{2}\right)$;
* if $\omega _{1}\cap \omega _{2}=\varnothing$ then $\pi \left(\omega _{1}\cup \omega _{2}\right)=\pi \left(\omega _{1}\right)+\pi \left(\omega _{2}\right)$;
If $\Omega$ is the $\sigma$-algebra of all Borels sets on a Hausdorff locally compact (or compact) space, then the following additional requirement is added:
* for every $x,y\in H,$ the map $\pi _{x,y}:\Omega \to \mathbb {C}$ 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 $x\in H,$ $\pi _{x,x}:\Omega \to \mathbb {C}$ is a positive measure on $\Omega$ with total variation $\left\|\pi _{x,x}\right\|=\pi _{x,x}(X)=\|x\|^{2}$ and that satisfies $\pi _{x,x}(\omega )=\langle \pi (\omega )x,x\rangle =\|\pi (\omega )x\|^{2}$ for all $\omega \in \Omega .$
For every $\omega _{1},\omega _{2}\in \Omega$:
* $\pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right)$ (since both are equal to $\pi \left(\omega _{1}\cap \omega _{2}\right)$).
* 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 $\pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=0=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right).$
* $\pi :\Omega \to {\mathcal {B}}(H)$ is finitely additive.
* If $\omega _{1},\omega _{2},\ldots$ are pairwise disjoint elements of $\Omega$ whose union is $\omega$ and if $\pi \left(\omega _{i}\right)=0$ for all $i$ then $\pi (\omega )=0.$
* For any fixed $x\in H,$ the map $\pi _{x}:\Omega \to H$ defined by $\pi _{x}(\omega ):=\pi (\omega )x$ is a countably additive $H$-valued measure on $\Omega .$
** Here countably additive means that whenever $\omega _{1},\omega _{2},\ldots$ are pairwise disjoint elements of $\Omega$ 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, $\sum _{i=1}^{\infty }\pi \left(\omega _{i}\right)x=\pi (\omega )x.$
** In other words, for every pairwise disjoint family of elements $\left(\omega _{i}\right)_{i=1}^{\infty }\subseteq \Omega$ whose union is $\omega _{\infty }\in \Omega$, 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 $\pi \left(\omega _{\infty }\right)$ in the strong operator topology on ${\mathcal {B}}(H)$: for every $x\in H$, the sequence of elements $\sum _{i=1}^{n}\pi \left(\omega _{i}\right)x$ converges to $\pi \left(\omega _{\infty }\right)x$ in $H$ (with respect to the norm topology).

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

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

### Essentially bounded functions

Suppose $f:X\to \mathbb {C}$ is a complex-valued $\Omega$-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 $D_{1},D_{2},\ldots$ be a basis for $\mathbb {C}$'s topology consisting of open disks and suppose that $D_{i_{1}},D_{i_{2}},\ldots$ 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 $D\cap \operatorname {Im} f=\varnothing$ then $\pi \left(f^{-1}(D)\right)=\pi (\varnothing )=0$ so that $D\subseteq V_{f}$ (although there are other ways in which $\pi \left(f^{-1}(D)\right)$ may equal 0). Indeed, $\mathbb {C} \setminus \operatorname {cl} (\operatorname {Im} f)\subseteq V_{f}.$
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 $\omega \in \Omega$ 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 $\|f\|^{\infty },$ to be the supremum of all $|\lambda |$ as $\lambda$ ranges over the essential range of $f.$

### Space of essentially bounded functions

Let ${\mathcal {B}}(X,\Omega )$ be the vector space of all bounded complex-valued $\Omega$-measurable functions $f:X\to \mathbb {C} ,$ which becomes a Banach algebra when normed by $\|f\|_{\infty }:=\sup _{x\in X}|f(x)|.$ The function $\|\,\cdot \,\|^{\infty }$ is a seminorm on ${\mathcal {B}}(X,\Omega ),$ but not necessarily a norm. The kernel of this seminorm, $N^{\infty }:=\left\{f\in {\mathcal {B}}(X,\Omega ):\|f\|^{\infty }=0\right\},$ is a vector subspace of ${\mathcal {B}}(X,\Omega )$ that is a closed two-sided ideal of the Banach algebra $\left({\mathcal {B}}(X,\Omega ),\|\cdot \|_{\infty }\right).$ Hence the quotient of ${\mathcal {B}}(X,\Omega )$ by $N^{\infty }$ is also a Banach algebra, denoted by $L^{\infty }(\pi ):={\mathcal {B}}(X,\Omega )/N^{\infty }$ where the norm of any element $f+N^{\infty }\in L^{\infty }(\pi )$ is equal to $\|f\|^{\infty }$ (since if $f+N^{\infty }=g+N^{\infty }$ then $\|f\|^{\infty }=\|g\|^{\infty }$) and this norm makes $L^{\infty }(\pi )$ into a Banach algebra. The spectrum of $f+N^{\infty }$ in $L^{\infty }(\pi )$ is the essential range of $f.$ This article will follow the usual practice of writing $f$ rather than $f+N^{\infty }$ to represent elements of $L^{\infty }(\pi ).$
Theorem — Let $\pi :\Omega \to {\mathcal {B}}(H)$ be a resolution of identity on $(X,\Omega ).$ There exists a closed normal subalgebra $A$ of ${\mathcal {B}}(H)$ and an isometric $^*$-isomorphism $\Psi :L^{\infty }(\pi )\to A$ satisfying the following properties:
* $\langle \Psi (f)x,y\rangle =\int _{X}f\operatorname {d} \pi _{x,y}$ for all $x,y\in H$ and $f\in L^{\infty }(\pi ),$ which justifies the notation $\Psi (f)=\int _{X}f\operatorname {d} \pi$;
* $\|\Psi (f)x\|^{2}=\int _{X}|f|^{2}\operatorname {d} \pi _{x,x}$ for all $x\in H$ and $f\in L^{\infty }(\pi )$;
* an operator $R\in \mathbb {B} (H)$ commutes with every element of $\operatorname {Im} \pi$ if and only if it commutes with every element of $A=\operatorname {Im} \Psi .$
* if $f$ is a simple function equal to $f=\sum _{i=1}^{n}\lambda _{i}\mathbb {1} _{\omega _{i}},$ where $\omega _{1},\ldots \omega _{n}$ is a partition of $X$ and the $\lambda _{i}$ are complex numbers, then $\Psi (f)=\sum _{i=1}^{n}\lambda _{i}\pi \left(\omega _{i}\right)$ (here $\mathbb {1}$ is the characteristic function);
* if $f$ is the limit (in the norm of $L^{\infty }(\pi )$) of a sequence of simple functions $s_{1},s_{2},\ldots$ in $L^{\infty }(\pi )$ then $\left(\Psi \left(s_{i}\right)\right)_{i=1}^{\infty }$ converges to$\Psi (f)$ in ${\mathcal {B}}(H)$ and $\|\Psi (f)\|=\|f\|^{\infty }$;
* $\left(\|f\|^{\infty }\right)^{2}=\sup _{\|h\|\leq 1}\int _{X}\operatorname {d} \pi _{h,h}$ for every $f\in L^{\infty }(\pi ).$

## 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 $\max \left\{|G(T)(h)|:h\in \sigma _{A}\right\},$ which is $\leq \|T\|.$
Theorem — Suppose $A$ is a closed normal subalgebra of ${\mathcal {B}}(H)$ that contains the identity operator $\operatorname {Id} _{H}$ and let $\sigma =\sigma _{A}$ be the maximal ideal space of $A.$ Let $\Omega$ 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 $\pi :\Omega \to A$ that satisfies: $\langle Tx,y\rangle =\int _{\sigma _{A}}G(T)\operatorname {d} \pi _{x,y}\quad {\text{ for all }}x,y\in H{\text{ and all }}T\in A;$ the notation $T=\int _{\sigma _{A}}G(T)\operatorname {d} \pi$ is used to summarize this situation. Let $I:\operatorname {Im} G\to A$ be the inverse of the Gelfand transform $G:A\to C\left(\sigma _{A}\right)$ where $\operatorname {Im} G$ can be canonically identified as a subspace of $L^{\infty }(\pi ).$ Let $B$ be the closure (in the norm topology of ${\mathcal {B}}(H)$) of the linear span of $\operatorname {Im} \pi .$ Then the following are true:
* $B$ is a closed subalgebra of ${\mathcal {B}}(H)$ containing $A.$
* There exists a (linear multiplicative) isometric $^*$-isomorphism $\Phi :L^{\infty }(\pi )\to B$ extending $I:\operatorname {Im} G\to A$ such that $\Phi f=\int _{\sigma _{A}}f\operatorname {d} \pi$ for all $f\in L^{\infty }(\pi ).$
** Recall that the notation $\Phi f=\int _{\sigma _{A}}f\operatorname {d} \pi$ means that $\langle (\Phi f)x,y\rangle =\int _{\sigma _{A}}f\operatorname {d} \pi _{x,y}$ for all $x,y\in H$;
** Note in particular that $T=\int _{\sigma _{A}}G(T)\operatorname {d} \pi =\Phi (G(T))$ for all $T\in A.$
** Explicitly, $\Phi$ satisfies $\Phi \left({\overline {f}}\right)=(\Phi f)^{*}$ and $\|\Phi f\|=\|f\|^{\infty }$ for every $f\in L^{\infty }(\pi )$ (so if $f$ is real valued then $\Phi (f)$ is self-adjoint).
* If $\omega \subseteq \sigma _{A}$ is open and nonempty (which implies that $\omega \in \Omega$) then $\pi (\omega )\neq 0.$
* A bounded linear operator $S\in {\mathcal {B}}(H)$ commutes with every element of $A$ if and only if it commutes with every element of $\operatorname {Im} \pi .$
The above result can be specialized to a single normal bounded operator.