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 ${\mathcal {B}}(H).$

Resolution of identity

See also: Projection-valued measure

Throughout, H is a fixed Hilbert space.

A projection-valued measure on a measurable space $\left(X,\Omega \right),$ 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 (i.e. $\pi \left(\omega \right)$ is a bounded linear operator $\pi \left(\omega \right):H\to H$ that satisfies $\pi \left(\omega \right)=\pi (\omega )^{*}$ 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 and y in H, the function $\Omega \to \mathbb {C}$ defined by $\omega \mapsto \langle \pi (\omega )x,t\rangle$ is a complex measure on $M$ (that is, a complex-valued countably additive function).

A resolution of identity on a measurable space $\left(X,\Omega \right)$ is a function $\pi :\Omega \to {\mathcal {B}}(H)$ such that for every $\omega _{1},\omega _{2}\in \Omega$:

1. $\pi \left(\varnothing \right)=0$;
2. $\pi (X)=\operatorname {Id} _{H}$;
3. for every $\omega \in \Omega ,$ $\pi \left(\omega \right)$ is a self-adjoint projection on H;
4. for every x and y in H, the map $\pi _{x,y}:\Omega \to \mathbb {C}$ defined by $\pi _{x,y}(\omega )=\left\langle \pi (\omega )x,y\right\rangle$ is a complex measure on $\Omega$;
5. $\pi \left(\omega _{1}\cap \omega _{2}\right)=\pi \left(\omega _{1}\right)\circ \pi \left(\omega _{2}\right)$;
6. 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:

7. for every x and 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 $\left\|\pi _{x,x}\right\|=\pi _{x,x}\left(X\right)=\|x\|^{2}$ and that satisfies $\pi _{x,x}\left(\omega \right)=\left\langle \pi (\omega )x,x\right\rangle =\|\pi \left(\omega \right)x\|^{2}$ 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 $\pi \left(\omega \right)=0.$
• However, $\pi :\Omega \to {\mathcal {B}}(H)$ is countably additive only in trivial situations as is now described: suppose that $\omega _{1},\omega _{2},\ldots$ are pairwise disjoint elements of $\Omega$ whose union is $\omega$ and that the partial sums $\sum _{i=1}^{n}\pi \left(\omega _{i}\right)$ converge to $\pi (\omega )$ in ${\mathcal {B}}(H)$ (with its norm topology) as $n\to \infty$; then since the norm of any projection is either 0 or $\geq 1,$ 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 $\pi _{x}:\Omega \to H$ 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 $\pi (\omega )$ in H. Said more succinctly, $\sum _{i=1}^{\infty }\pi \left(\omega _{i}\right)x=\pi \left(\omega \right)x.$

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

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

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 $\pi \left(f^{-1}\left(D\right)\right)=\pi \left(\varnothing \right)=0$ so that $D\subseteq V_{f}$ (although there are other ways in which $\pi \left(f^{-1}\left(D\right)\right)$ may equal 0). Indeed, $\mathbb {C} \setminus \operatorname {cl} \left(\operatorname {Im} f\right)\subseteq V_{f}.$

The essential range of f is defined to be the complement of $V_{f}.$ It is the smallest closed subset of $\mathbb {C}$ that contains $f(x)$ for almost all $x\in X$ (i.e. for all $x\in X$ except for those in some set $\omega \in \Omega$ such that $\pi \left(\omega \right)=0$). 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 $\left\|\cdot \right\|^{\infty }$ is a seminorm on ${\mathcal {B}}(X,\Omega ),$ but not necessarily a norm. The kernel of this seminorm, $N^{\infty }:=\left\{f\in {\mathcal {B}}(X,\Omega ):\left\|f\right\|^{\infty }=0\right\},$ 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 $\left(X,\Omega \right).$ 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:

1. $\left\langle \Psi (f)x,y\right\rangle =\int _{X}f\operatorname {d} \pi _{x,y}$ for all x and y in H and $f\in L^{\infty }\left(\pi \right),$ which justifies the notation $\Psi (f)=\int _{X}f\operatorname {d} \pi$;
2. $\left\|\Psi (f)x\right\|^{2}=\int _{X}|f|^{2}\operatorname {d} \pi _{x,x}$ for all $x\in H$ and $f\in L^{\infty }\left(\pi \right)$;
3. 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 $A=\operatorname {Im} \Psi .$
4. 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);
5. 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 $\|\Psi (f)\|=\|f\|^{\infty }$;
6. $\left(\|f\|^{\infty }\right)^{2}=\sup _{\|h\|\leq 1}\int _{X}\operatorname {d} \pi _{h,h}$ for every $f\in L^{\infty }\left(\pi \right).$

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:

$\left\langle Tx,y\right\rangle =\int _{\sigma _{A}}G(T)\operatorname {d} \pi _{x,y}$ for all $x,y\in H$ 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:

1. B is a closed subalgebra of ${\mathcal {B}}(H)$ containing A;
2. There exists a (linear multiplicative) isometric ^*-isomorphism $\Phi :L^{\infty }\left(\pi \right)\to B$ 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 $\left\langle \left(\Phi f\right)x,y\right\rangle =\int _{\sigma _{A}}f\operatorname {d} \pi _{x,y}$ for all $x,y\in H$;
   • Note in particular that $T=\int _{\sigma _{A}}G(T)\operatorname {d} \pi =\Phi \left(G(T)\right)$ for all $T\in A$;
   • Explicitly, $\Phi$ satisfies $\Phi \left({\overline {f}}\right)=\left(\Phi f\right)^{*}$ and $\|\Phi f\|=\|f\|^{\infty }$ for every $f\in L^{\infty }(\pi )$ (so if f is real valued then $\Phi (f)$ is self-adjoint);
3. If $\omega \subseteq \sigma _{A}$ is open and nonempty (which implies that $\omega \in \Omega$) then $\pi \left(\omega \right)\neq 0$;
4. 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.

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. {{cite book}}: Invalid |ref=harv (help)
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.