Spectral theory of normal C*-algebras explained

In functional analysis, every C^*-algebra is isomorphic to a subalgebra of the C^*-algebra

l{B}(H) of bounded linear operators on some Hilbert space H. This article describes the spectral theory of closed normal subalgebras of l{B}(H) . A subalgebra A of l{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

See also: Projection-valued measure.

Throughout,

H is a fixed Hilbert space.(X, \Omega), where \Omega is a σ-algebra of subsets of X, is a mapping \pi : \Omega \to l{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$$\backslash pi(X)\; =\; \backslash operatorname\_H\; \backslash quad$$(where \operatorname{Id} _{H} is the identity operator of H ) and for every x, y \in H, the function \Omega \to \Complex defined by \omega \mapsto \langle \pi(\omega)x, y \rangle is a complex measure on M (that is, a complex-valued countably additive function).(X, \Omega) is a function \pi : \Omega \to l{B}(H) such that for every \omega _{1,} \omega _{2} \in \Omega :

1. \pi(\varnothing) = 0 ;
2. \pi(X) = \operatorname{Id} _{H} ;
3. for every \omega \in \Omega, \pi(\omega) is a self-adjoint projection on H ;
4. for every x, y \in H, the map \pi _{x,} : \Omega \to \Complex defined by \pi _{x,y} (\omega) = \langle \pi(\omega) x, y \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, y \in H, the map \pi _{x,} : \Omega \to \Complex 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,} : \Omega \to \Complex is a positive measure on \Omega with total variation \left\|\pi _{x,} \right\| = \pi _{x,} (X) = \|x\| ^{2} and that satisfies \pi _{x,} (\omega) = \langle \pi(\omega) x, x \rangle = \|\pi(\omega) 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 l{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(\omega) = 0 .
• However,
\pi : \Omega \to l{B}(H) is 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 ^{n} _{i =1} \pi\left(\omega _{i\right)} converge to \pi(\omega) in l{B}(H) (with its norm topology) as n \to inf ty ; 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 ^{n} _{i =1} \pi\left(\omega _{i\right)} x converge to \pi(\omega)x in H. Said more succinctly, \sum ^{inf ty} _{i =1} \pi\left(\omega _{i\right)} x = \pi(\omega) x.
  • In other words, for every pairwise disjoint family of elements
  \left(\omega _{i\right)} ^{inf ty} ^{i =1} \subseteq \Omega whose union is \omega _{infty} \in\Omega , then \sum ^{n} _{i =1} \pi\left(\omega _{i\right)} = \pi\left( c up ^{n} _{i =1} \omega _{i\right)} (by finite additivity of \pi ) converges to \pi\left(\omega _{infty} \right) in the strong operator topology on l{B}(H) : for every x\in H , the sequence of elements \sum ^{n} _{i =1} \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 l{B}(H) be a resolution of identity on (X, \Omega).

Essentially bounded functions

Suppose

f : X \to \Complex is a complex-valued \Omega -measurable function. There exists a unique largest open subset V _{f} of \Complex (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 \Complex '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 … = V _{f.} Note that, in particular, if D is an open subset of \Complex 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). Indeed, \Complex \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 \Complex 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 ). The essential range is a closed subset of \Complex so that if it is also a bounded subset of \Complex 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 | λ | as λ ranges over the essential range of f.

Space of essentially bounded functions

Let

l{B}(X, \Omega) be the vector space of all bounded complex-valued \Omega -measurable functions f : X \to \Complex, which becomes a Banach algebra when normed by \|f\| _{infty} : = \sup _{x} |f(x) |. The function \| ⋅ \| ^{infty} is a seminorm on l{B}(X, \Omega), but not necessarily a norm. The kernel of this seminorm, N ^{infty} : = \left\{ f \in l{B}(X, \Omega) : \|f\| ^{infty} = 0 \right\}, is a vector subspace of l{B}(X, \Omega) that is a closed two-sided ideal of the Banach algebra \left( l{B}(X, \Omega), \| ⋅ \| _{infty} \right). Hence the quotient of l{B}(X, \Omega) by N ^{infty} is also a Banach algebra, denoted by L ^{infty} (\pi) : = l{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).

Spectral theorem

The maximal ideal space of a Banach algebra

A is the set of all complex homomorphisms A \to \Complex, 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 \Complex defined by G(T)(h) : = h(T). \sigma _{A} is given the weakest topology making every G(T) : \sigma _{A} \to \Complex 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\|.

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

References

• Book: Robertson, A. P.. Topological vector spaces. University Press. Cambridge England. 1973. 0-521-29882-2. 589250 .

Notes and References

1. Book: Rudin, Walter . Functional Analysis . McGraw Hill . 1991 . 0-07-100944-2 . 2nd . New York . 292–293 . en.