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 $\pi (\omega )=\pi (\omega {)}^{\ast }$ 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 $\mathrm{\Omega }\to ℂ$ 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 identityTemplate:Sfn 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 ${\pi }_{x,y}:\mathrm{\Omega }\to ℂ$ 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 ${\pi }_{x,y}:\mathrm{\Omega }\to ℂ$ 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 $x\in H,$ ${\pi }_{x,x}:\mathrm{\Omega }\to ℂ$ 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 $\omega \in \mathrm{\Omega }.$Template:Sfn

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 $\pi ({\omega }_1\cap {\omega }_2)$).Template:Sfn
• 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 $\pi \left({\omega }_1\right)\pi \left({\omega }_2\right)=0=\pi \left({\omega }_2\right)\pi \left({\omega }_1\right).$Template:Sfn
• $\pi :\mathrm{\Omega }\to ℬ(H)$ is finitely additive.Template:Sfn
• 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 $\pi (\omega )=0.$Template:Sfn
• However, $\pi :\mathrm{\Omega }\to ℬ(H)$ is Template:Em 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 $0.$Template:Sfn
• 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, $\sum _{i=1}^{\mathrm{\infty }}\pi \left({\omega }_i\right)x=\pi (\omega )x.$Template:Sfn
  • 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 $f:X\to ℂ$ is a complex-valued $\mathrm{\Omega }$-measurable function. There exists a unique largest open subset $V_f$ of $ℂ$ (ordered under subset inclusion) such that $\pi \left(f^{-1}\left(V_f\right)\right)=0.$Template:Sfn To see why, let $D_1,D_2,\dots$ be a basis for $ℂ$'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 $ℂ$ 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 Template:Math). Indeed, $ℂ\setminus \mathrm{cl}(\mathrm{Im}f)\subseteq V_f.$

The essential range of $f$ is defined to be the complement of $V_f.$ It is the smallest closed subset of $ℂ$ 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 $\pi (\omega )=0$).Template:Sfn The essential range is a closed subset of $ℂ$ so that if it is also a bounded subset of $ℂ$ 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 $f.$Template:Sfn

Space of essentially bounded functions

Let $ℬ(X,\mathrm{\Omega })$ be the vector space of all bounded complex-valued $\mathrm{\Omega }$-measurable functions $f:X\to ℂ,$ 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 $(ℬ(X,\mathrm{\Omega }),\Vert \cdot {\Vert }_{\mathrm{\infty }}).$Template:Sfn 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 $f.$Template:Sfn This article will follow the usual practice of writing $f$ rather than $f+N^{\mathrm{\infty }}$ to represent elements of $L^{\mathrm{\infty }}(\pi ).$

Spectral theorem

The maximal ideal space of a Banach algebra $A$ is the set of all complex homomorphisms $A\to ℂ,$ which we'll denote by ${\sigma }_A.$ For every $T$ in $A,$ the Gelfand transform of $T$ is the map $G(T):{\sigma }_A\to ℂ$ defined by $G(T)(h):=h(T).$ ${\sigma }_A$ is given the weakest topology making every $G(T):{\sigma }_A\to ℂ$ 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 $\le \Vert T\Vert .$Template:Sfn

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

See also

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References

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• Template:Robertson Topological Vector Spaces
• Template:Rudin Walter Functional Analysis
• Template:Schaefer Wolff Topological Vector Spaces