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In functional analysis, every C-algebra is isomorphic to a subalgebra of the C-algebra ${\displaystyle {\mathcal {B}}(H)}$ of bounded linear operators on some Hilbert space ${\displaystyle H.}$ This article describes the spectral theory of closed normal subalgebras of ${\displaystyle {\mathcal {B}}(H)}$. A subalgebra ${\displaystyle A}$ of ${\displaystyle {\mathcal {B}}(H)}$ is called normal if it is commutative and closed under the ${\displaystyle \ast }$ operation: for all ${\displaystyle x,y\in A}$, we have ${\displaystyle x^{\ast }\in A}$ and that ${\displaystyle xy=yx}$.

Resolution of identity

Throughout, ${\displaystyle H}$ is a fixed Hilbert space.

A projection-valued measure on a measurable space ${\displaystyle (X,\Omega ),}$ where ${\displaystyle \Omega }$ is a σ-algebra of subsets of ${\displaystyle X,}$ is a mapping ${\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}$ such that for all ${\displaystyle \omega \in \Omega ,}$ ${\displaystyle \pi (\omega )}$ is a self-adjoint projection on ${\displaystyle H}$ (that is, ${\displaystyle \pi (\omega )}$ is a bounded linear operator ${\displaystyle \pi (\omega ):H\to H}$ that satisfies ${\displaystyle \pi (\omega )=\pi (\omega )^{*}}$ and ${\displaystyle \pi (\omega )\circ \pi (\omega )=\pi (\omega )}$) such that $${\displaystyle \pi (X)=\operatorname {Id} _{H}\quad }$$ (where ${\displaystyle \operatorname {Id} _{H}}$ is the identity operator of ${\displaystyle H}$) and for every ${\displaystyle x,y\in H,}$ the function ${\displaystyle \Omega \to \mathbb {C} }$ defined by ${\displaystyle \omega \mapsto \langle \pi (\omega )x,y\rangle }$ is a complex measure on ${\displaystyle M}$ (that is, a complex-valued countably additive function).

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

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

If ${\displaystyle \Omega }$ is the ${\displaystyle \sigma }$-algebra of all Borels sets on a Hausdorff locally compact (or compact) space, then the following additional requirement is added:

7. for every ${\displaystyle x,y\in H,}$ the map ${\displaystyle \pi _{x,y}:\Omega \to \mathbb {C} }$ is a regular Borel measure (this is automatically satisfied on compact metric spaces).

Conditions 2, 3, and 4 imply that ${\displaystyle \pi }$ is a projection-valued measure.

Properties

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

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 ${\displaystyle \pi \left(\omega _{1}\cap \omega _{2}\right)}$).
• If ${\displaystyle \omega _{1}\cap \omega _{2}=\varnothing }$ 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 ${\displaystyle \pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=0=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right).}$
• ${\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}$ is finitely additive.
• 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 ${\displaystyle i}$ then ${\displaystyle \pi (\omega )=0.}$
• However, ${\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}$ is countably additive only in trivial situations as is now described: 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 ${\displaystyle 0}$ 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 ${\displaystyle 0.}$
• For any fixed ${\displaystyle x\in H,}$ the map ${\displaystyle \pi _{x}:\Omega \to H}$ defined by ${\displaystyle \pi _{x}(\omega ):=\pi (\omega )x}$ is a countably additive ${\displaystyle H}$-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 ${\displaystyle \pi (\omega )x}$ in ${\displaystyle H.}$ Said more succinctly, ${\displaystyle \sum _{i=1}^{\infty }\pi \left(\omega _{i}\right)x=\pi (\omega )x.}$
  • In other words, for every pairwise disjoint family of elements ${\displaystyle \left(\omega _{i}\right)_{i=1}^{\infty }\subseteq \Omega }$ whose union is ${\displaystyle \omega _{\infty }\in \Omega }$, then ${\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)=\pi \left(\bigcup _{i=1}^{n}\omega _{i}\right)}$ (by finite additivity of ${\displaystyle \pi }$) converges to ${\displaystyle \pi \left(\omega _{\infty }\right)}$ in the strong operator topology on ${\displaystyle {\mathcal {B}}(H)}$: for every ${\displaystyle x\in H}$, the sequence of elements ${\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)x}$ converges to ${\displaystyle \pi \left(\omega _{\infty }\right)x}$ in ${\displaystyle H}$ (with respect to the norm topology).

L(π) - space of essentially bounded function

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

Essentially bounded functions

Suppose ${\displaystyle f:X\to \mathbb {C} }$ is a complex-valued ${\displaystyle \Omega }$-measurable function. There exists a unique largest open subset ${\displaystyle V_{f}}$ of ${\displaystyle \mathbb {C} }$ (ordered under subset inclusion) such that ${\displaystyle \pi \left(f^{-1}\left(V_{f}\right)\right)=0.}$ 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 ${\displaystyle D}$ is an open subset of ${\displaystyle \mathbb {C} }$ such that ${\displaystyle D\cap \operatorname {Im} f=\varnothing }$ then ${\displaystyle \pi \left(f^{-1}(D)\right)=\pi (\varnothing )=0}$ so that ${\displaystyle D\subseteq V_{f}}$ (although there are other ways in which ${\displaystyle \pi \left(f^{-1}(D)\right)}$ may equal 0). Indeed, ${\displaystyle \mathbb {C} \setminus \operatorname {cl} (\operatorname {Im} f)\subseteq V_{f}.}$

The essential range of ${\displaystyle f}$ 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 ${\displaystyle x\in X}$ (that is, for all ${\displaystyle x\in X}$ except for those in some set ${\displaystyle \omega \in \Omega }$ such that ${\displaystyle \pi (\omega )=0}$). 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.

The function ${\displaystyle f}$ is essentially bounded if its essential range is bounded, in which case 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 ${\displaystyle f.}$

Space of essentially bounded functions

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 ${\displaystyle \|f\|_{\infty }:=\sup _{x\in X}|f(x)|.}$ The function ${\displaystyle \|\,\cdot \,\|^{\infty }}$ is a seminorm on ${\displaystyle {\mathcal {B}}(X,\Omega ),}$ but not necessarily a norm. The kernel of this seminorm, ${\displaystyle N^{\infty }:=\left\{f\in {\mathcal {B}}(X,\Omega ):\|f\|^{\infty }=0\right\},}$ is a vector subspace of ${\displaystyle {\mathcal {B}}(X,\Omega )}$ that is a closed two-sided ideal of the Banach algebra ${\displaystyle \left({\mathcal {B}}(X,\Omega ),\|\cdot \|_{\infty }\right).}$ 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 ${\displaystyle f.}$ This article will follow the usual practice of writing ${\displaystyle f}$ rather than ${\displaystyle f+N^{\infty }}$ to represent elements of ${\displaystyle L^{\infty }(\pi ).}$

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

1. ${\displaystyle \langle \Psi (f)x,y\rangle =\int _{X}f\operatorname {d} \pi _{x,y}}$ for all ${\displaystyle x,y\in H}$ and ${\displaystyle f\in L^{\infty }(\pi ),}$ which justifies the notation ${\displaystyle \Psi (f)=\int _{X}f\operatorname {d} \pi }$;
2. ${\displaystyle \|\Psi (f)x\|^{2}=\int _{X}|f|^{2}\operatorname {d} \pi _{x,x}}$ for all ${\displaystyle x\in H}$ and ${\displaystyle f\in L^{\infty }(\pi )}$;
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 ${\displaystyle f}$ 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 ${\displaystyle X}$ 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 ${\displaystyle f}$ 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 ${\displaystyle f\in L^{\infty }(\pi ).}$

Spectral theorem

The maximal ideal space of a Banach algebra ${\displaystyle A}$ is the set of all complex homomorphisms ${\displaystyle A\to \mathbb {C} ,}$ which we'll denote by ${\displaystyle \sigma _{A}.}$ For every ${\displaystyle T}$ in ${\displaystyle A,}$ the Gelfand transform of ${\displaystyle T}$ is the map ${\displaystyle G(T):\sigma _{A}\to \mathbb {C} }$ defined by ${\displaystyle G(T)(h):=h(T).}$ ${\displaystyle \sigma _{A}}$ is given the weakest topology making every ${\displaystyle G(T):\sigma _{A}\to \mathbb {C} }$ continuous. With this topology, ${\displaystyle \sigma _{A}}$ is a compact Hausdorff space and every ${\displaystyle T}$ in ${\displaystyle A,}$ ${\displaystyle G(T)}$ belongs to ${\displaystyle C\left(\sigma _{A}\right),}$ which is the space of continuous complex-valued functions on ${\displaystyle \sigma _{A}.}$ The range of ${\displaystyle G(T)}$ is the spectrum ${\displaystyle \sigma (T)}$ and that the spectral radius is equal to ${\displaystyle \max \left\{|G(T)(h)|:h\in \sigma _{A}\right\},}$ which is ${\displaystyle \leq \|T\|.}$

Theorem — Suppose ${\displaystyle A}$ is a closed normal subalgebra of ${\displaystyle {\mathcal {B}}(H)}$ that contains the identity operator ${\displaystyle \operatorname {Id} _{H}}$ and let ${\displaystyle \sigma =\sigma _{A}}$ be the maximal ideal space of ${\displaystyle A.}$ Let ${\displaystyle \Omega }$ be the Borel subsets of ${\displaystyle \sigma .}$ For every ${\displaystyle T}$ in ${\displaystyle A,}$ let ${\displaystyle G(T):\sigma _{A}\to \mathbb {C} }$ denote the Gelfand transform of ${\displaystyle T}$ so that ${\displaystyle G}$ is an injective map ${\displaystyle G:A\to C\left(\sigma _{A}\right).}$ There exists a unique resolution of identity ${\displaystyle \pi :\Omega \to A}$ that satisfies: $${\displaystyle \langle Tx,y\rangle =\int _{\sigma _{A}}G(T)\operatorname {d} \pi _{x,y}\quad {\text{ for all }}x,y\in H{\text{ and all }}T\in A;}$$ the notation ${\displaystyle T=\int _{\sigma _{A}}G(T)\operatorname {d} \pi }$ is used to summarize this situation. Let ${\displaystyle I:\operatorname {Im} G\to A}$ be the inverse of the Gelfand transform ${\displaystyle G:A\to C\left(\sigma _{A}\right)}$ where ${\displaystyle \operatorname {Im} G}$ can be canonically identified as a subspace of ${\displaystyle L^{\infty }(\pi ).}$ Let ${\displaystyle B}$ be the closure (in the norm topology of ${\displaystyle {\mathcal {B}}(H)}$) of the linear span of ${\displaystyle \operatorname {Im} \pi .}$ Then the following are true:

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

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

1. Rudin, Walter (1991). Functional Analysis (2nd ed.). New York: McGraw Hill. pp. 292–293. ISBN 0-07-100944-2.
2. ^ Rudin 1991, pp. 316–318.
3. ^ Rudin 1991, pp. 318–321.
4. Rudin 1991, p. 280.
5. 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.

• C*-algebras
• Spectral theory