Spectral theory of normal C*-algebras

Throughout let ${\displaystyle \pi }$ be a resolution of identity.

• For all x in H, ${\displaystyle \pi _{x,x}:\Omega \to \mathbb {C} }$ is a positive measure on ${\displaystyle \Omega }$ with total variation ${\displaystyle \left\|\pi _{x,x}\right\|=\pi _{x,x}\left(X\right)=\|x\|^{2}}$ and that satisfies ${\displaystyle \pi _{x,x}\left(\omega \right)=\left\langle \pi (\omega )x,x\right\rangle =\|\pi \left(\omega \right)x\|^{2}}$ for all ${\displaystyle \omega \in \Omega }$.[1]

For every ${\displaystyle \omega _{1},\omega _{2}\in \Omega }$:

• ${\displaystyle \pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right)}$ (since both are equal to ${\displaystyle \pi \left(\omega _{1}\cap \omega _{2}\right)}$).[1]
• If ${\displaystyle \omega _{1}\cap \omega _{2}=\emptyset }$ then the ranges of the maps ${\displaystyle \pi \left(\omega _{1}\right)}$ and ${\displaystyle \pi \left(\omega _{2}\right)}$ are orthogonal to each other and ${\displaystyle \pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=0=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right)}$.[1]
• ${\displaystyle \pi$ :\Omega \to {\mathcal {B}}(H)} is finitely additive.[1]
• If ${\displaystyle \omega _{1},\omega _{2},\ldots }$ are pairwise disjoint elements of ${\displaystyle \Omega }$ whose union is ${\displaystyle \omega }$ and if ${\displaystyle \pi \left(\omega _{i}\right)=0}$ for all i then ${\displaystyle \pi \left(\omega \right)=0}$.[1]
• However, ${\displaystyle \pi$ :\Omega \to {\mathcal {B}}(H)} is countably additive only in trivial situations as we now describe: suppose that ${\displaystyle \omega _{1},\omega _{2},\ldots }$ are pairwise disjoint elements of ${\displaystyle \Omega }$ whose union is ${\displaystyle \omega }$ and that the partial sums ${\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)}$ converge to ${\displaystyle \pi (\omega )}$ in ${\displaystyle {\mathcal {B}}(H)}$ (with its norm topology) as ${\displaystyle n\to \infty }$; then since the norm of any projection is either 0 or ${\displaystyle \geq 1}$, the partial sums cannot form a Cauchy sequence unless all but finitely many of the ${\displaystyle \pi \left(\omega _{i}\right)}$ are 0.[1]
• For any fixed x in H, the map ${\displaystyle \pi _{x}:\Omega \to H}$ defined by ${\displaystyle \pi _{x}(\omega ):=\pi (\omega )x}$ is a countably additive H-valued measure on ${\displaystyle \Omega }$.
  • Here countably additive means that whenever ${\displaystyle \omega _{1},\omega _{2},\ldots }$ are pairwise disjoint elements of ${\displaystyle \Omega }$ whose union is ${\displaystyle \omega }$, then the partial sums ${\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)x}$ converge to ${\displaystyle \pi (\omega )}$ in H (or succinctly, ${\displaystyle \sum _{i=1}^{\infty }\pi \left(\omega _{i}\right)x=\pi \left(\omega \right)x}$).[1]

Suppose we are given a complex-valued ${\displaystyle \Omega }$-measurable functions ${\displaystyle f:X\to \mathbb {C} }$. There exists a unique largest open subsets ${\displaystyle W_{f}}$ of ${\displaystyle \mathbb {C} }$ (ordered under subset inclusion) such that ${\displaystyle \pi \left(f^{-1}\left(V_{f}\right)\right)=0}$.[2] To see why, let ${\displaystyle D_{1},D_{2},\ldots }$ be a basis for ${\displaystyle \mathbb {C} }$'s topology consisting of open disks and suppose that ${\displaystyle D_{i_{1}},D_{i_{2}},\ldots }$ is the subsequence (possibly finite) consisting of those sets such that ${\displaystyle \pi \left(f^{-1}\left(D_{i_{k}}\right)\right)=0}$; then ${\displaystyle D_{i_{1}}\cup D_{i_{2}}\cup \cdots =V_{f}}$. Note that, in particular, if D is an open subset of ${\displaystyle \mathbb {C} }$ such that ${\displaystyle D\cap \operatorname {Im} f=\emptyset }$ then ${\displaystyle \pi \left(f^{-1}\left(D\right)\right)=\pi \left(\emptyset \right)=0}$ so that ${\displaystyle D\subseteq V_{f}}$ (although there are other ways in which ${\displaystyle \pi \left(f^{-1}\left(D\right)\right)}$ may equal 0). Indeed, ${\displaystyle \mathbb {C} \setminus \operatorname {cl} \left(\operatorname {Im} f\right)\subseteq V_{f}}$.

The essential range of f is defined to be the complement of ${\displaystyle V_{f}}$. It is the smallest closed subset of ${\displaystyle \mathbb {C} }$ that contains ${\displaystyle f(x)}$ for almost all ${\displaystyle x\in X}$ (i.e. for all ${\displaystyle x\in X}$ except for those in some set ${\displaystyle \omega \in \Omega }$ such that ${\displaystyle \pi \left(\omega \right)=0}$).[2] Note that the essential range is a closed subset of ${\displaystyle \mathbb {C} }$ so that if it is also a bounded subset of ${\displaystyle \mathbb {C} }$ then it is compact.

We say that f is essentially bounded if its essential range is bounded, in which case we define its essential supremum, denoted by ${\displaystyle \|f\|^{\infty }}$, to be the supremum of all ${\displaystyle |\lambda |}$ as ${\displaystyle \lambda }$ ranges over the essential range of f.[2]

Let ${\displaystyle {\mathcal {B}}(X,\Omega )}$ be the vector space of all bounded complex-valued ${\displaystyle \Omega }$-measurable functions ${\displaystyle f:X\to \mathbb {C} }$, which becomes a Banach algebra when normed by ${\displaystyle \|f\|_{\infty }:=\sup _{x\in X}|f(x)|}$. Note that ${\displaystyle \left\|\cdot \right\|^{\infty }}$ is a seminorm on ${\displaystyle {\mathcal {B}}(X,\Omega )}$, but not necessarily a norm. The kernel of this seminorm, ${\displaystyle N^{\infty }:=\left\{f\in {\mathcal {B}}(X,\Omega ):\left\|f\right\|^{\infty }=0\right\}}$, is a vector subspace of ${\displaystyle {\mathcal {B}}(X,\Omega )}$ that is a closed two-sided ideal of the Banach algebra ${\displaystyle \left({\mathcal {B}}(X,\Omega ),\|\cdot \|_{\infty }\right)}$.[2] Hence the quotient of ${\displaystyle {\mathcal {B}}(X,\Omega )}$ by ${\displaystyle N^{\infty }}$ is also a Banach algebra, denoted by ${\displaystyle L^{\infty }(\pi ):={\mathcal {B}}(X,\Omega )/N^{\infty }}$ where the norm of any element ${\displaystyle f+N^{\infty }\in L^{\infty }(\pi )}$ is equal to ${\displaystyle \|f\|^{\infty }}$ (since if ${\displaystyle f+N^{\infty }=g+N^{\infty }}$ then ${\displaystyle \|f\|^{\infty }=\|g\|^{\infty }}$) and this norm makes ${\displaystyle L^{\infty }(\pi )}$ into a Banach algebra. The spectrum of ${\displaystyle f+N^{\infty }}$ in ${\displaystyle L^{\infty }(\pi )}$ is the essential range of f.[2] We will follow the usual practice of writing f rather than ${\displaystyle f+N^{\infty }}$ to represent elements of ${\displaystyle L^{\infty }(\pi )}$.

Theorem:[2] Let ${\displaystyle \pi$ :\Omega \to {\mathcal {B}}(H)} be a resolution of identity on ${\displaystyle \left(X,\Omega \right)}$. There exists a closed normal subalgebra A of ${\displaystyle {\mathcal {B}}(H)}$ and an isometric ^*-isomorphism ${\displaystyle \Psi :L^{\infty }(\pi )\to A}$ satisfying the following properties:

1. ${\displaystyle \left\langle \Psi (f)x,y\right\rangle =\int _{X}f\operatorname {d} \pi _{x,y}}$ for all x and y in H and ${\displaystyle f\in L^{\infty }\left(\pi \right)}$, which justifies the notation ${\displaystyle \Psi (f)=\int _{X}f\operatorname {d} \pi }$;
2. ${\displaystyle \left\|\Psi (f)x\right\|^{2}=\int _{X}|f|^{2}\operatorname {d} \pi _{x,x}}$ for all ${\displaystyle x\in H}$ and ${\displaystyle f\in L^{\infty }\left(\pi \right)}$;
3. an operator ${\displaystyle R\in \mathbb {B} (H)}$ commutes with every element of ${\displaystyle \operatorname {Im} \pi }$ if and only if it commutes with every element of ${\displaystyle A=\operatorname {Im} \Psi }$.
4. if f is a simple function equal to ${\displaystyle f=\sum _{i=1}^{n}\lambda _{i}\mathbb {1} _{\omega _{i}}}$, where ${\displaystyle \omega _{1},\ldots \omega _{n}}$ is a partition of X and the ${\displaystyle \lambda _{i}}$ are complex numbers, then ${\displaystyle \Psi (f)=\sum _{i=1}^{n}\lambda _{i}\pi \left(\omega _{i}\right)}$ (here ${\displaystyle \mathbb {1} }$ is the characteristic function);
5. if f is the limit (in the norm of ${\displaystyle L^{\infty }(\pi )}$) of a sequence of simple functions ${\displaystyle s_{1},s_{2},\ldots }$ in ${\displaystyle L^{\infty }(\pi )}$ then ${\displaystyle \left(\Psi \left(s_{i}\right)\right)_{i=1}^{\infty }}$ converges to${\displaystyle \Psi (f)}$ in ${\displaystyle {\mathcal {B}}(H)}$ and ${\displaystyle \|\Psi (f)\|=\|f\|^{\infty }}$;
6. ${\displaystyle \left(\|f\|^{\infty }\right)^{2}=\sup _{\|h\|\leq 1}\int _{X}\operatorname {d} \pi _{h,h}}$ for every ${\displaystyle f\in L^{\infty }\left(\pi \right)}$.