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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 $xy=yx$ .^[1]

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^[2] 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 $\omega \in \Omega .$ ^[2]

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 $\pi \left(\omega _{1}\cap \omega _{2}\right)$ ).^[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 $\pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=0=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right).$ ^[2]
$\pi :\Omega \to {\mathcal {B}}(H)$ is finitely additive.^[2]
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 $\pi (\omega )=0.$ ^[2]

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 $0.$ ^[2]

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, $\sum _{i=1}^{\infty }\pi \left(\omega _{i}\right)x=\pi (\omega )x.$ ^[2]
- 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 $\pi \left(f^{-1}\left(V_{f}\right)\right)=0.$ ^[3] 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 $\pi (\omega )=0$ ).^[3] 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 $f.$ ^[3]

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 $\left({\mathcal {B}}(X,\Omega ),\|\cdot \|_{\infty }\right).$ ^[3] 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 $f.$ ^[3] This article will follow the usual practice of writing $f$ rather than $f+N^{\infty }$ to represent elements of $L^{\infty }(\pi ).$

Theorem^[3] — 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 $\leq \|T\|.$ ^[4]

Theorem^[5] — 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.

References

^ Rudin, Walter (1991). Functional Analysis (2nd ed.). New York: McGraw Hill. pp. 292–293. ISBN 0-07-100944-2.
^ ^a ^b ^c ^d ^e ^f ^g ^h Rudin 1991, pp. 316–318.
^ ^a ^b ^c ^d ^e ^f Rudin 1991, pp. 318–321.
^ Rudin 1991, p. 280.
^ Rudin 1991, pp. 321–325.

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. 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.

[1] Rudin, Walter (1991). Functional Analysis (2nd ed.). New York: McGraw Hill. pp. 292–293. ISBN 0-07-100944-2.

[FOOTNOTERudin1991316–318-2] ^ ^a ^b ^c ^d ^e ^f ^g ^h Rudin 1991, pp. 316–318.

[FOOTNOTERudin1991318–321-3] ^ ^a ^b ^c ^d ^e ^f Rudin 1991, pp. 318–321.

[FOOTNOTERudin1991280-4] Rudin 1991, p. 280.

[FOOTNOTERudin1991321–325-5] Rudin 1991, pp. 321–325.

[1]

[2]

[3]

[4]

[5]

Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem