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Spectral theory of normal C*-algebras

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In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra of bounded linear operators on some Hilbert space This article describes the spectral theory of closed normal subalgebras of . A subalgebra of is called normal if it is commutative and closed under the operation: for all , we have and that .[1]

Resolution of identity

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Throughout, is a fixed Hilbert space.

A projection-valued measure on a measurable space where is a σ-algebra of subsets of is a mapping such that for all is a self-adjoint projection on (that is, is a bounded linear operator that satisfies and ) such that (where is the identity operator of ) and for every the function defined by is a complex measure on (that is, a complex-valued countably additive function).

A resolution of identity[2] on a measurable space is a function such that for every :

  1. ;
  2. ;
  3. for every is a self-adjoint projection on ;
  4. for every the map defined by is a complex measure on ;
  5. ;
  6. if then ;

If is the -algebra of all Borels sets on a Hausdorff locally compact (or compact) space, then the following additional requirement is added:

  1. for every the map is a regular Borel measure (this is automatically satisfied on compact metric spaces).

Conditions 2, 3, and 4 imply that is a projection-valued measure.

Properties

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Throughout, let be a resolution of identity. For all is a positive measure on with total variation and that satisfies for all [2]

For every :

  • (since both are equal to ).[2]
  • If then the ranges of the maps and are orthogonal to each other and [2]
  • is finitely additive.[2]
  • If are pairwise disjoint elements of whose union is and if for all then [2]
    • However, is countably additive only in trivial situations as is now described: suppose that are pairwise disjoint elements of whose union is and that the partial sums converge to in (with its norm topology) as ; then since the norm of any projection is either or the partial sums cannot form a Cauchy sequence unless all but finitely many of the are [2]
  • For any fixed the map defined by is a countably additive -valued measure on
    • Here countably additive means that whenever are pairwise disjoint elements of whose union is then the partial sums converge to in Said more succinctly, [2]
    • In other words, for every pairwise disjoint family of elements whose union is , then (by finite additivity of ) converges to in the strong operator topology on : for every , the sequence of elements converges to in (with respect to the norm topology).

L(π) - space of essentially bounded function

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The be a resolution of identity on

Essentially bounded functions

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Suppose is a complex-valued -measurable function. There exists a unique largest open subset of (ordered under subset inclusion) such that [3] To see why, let be a basis for 's topology consisting of open disks and suppose that is the subsequence (possibly finite) consisting of those sets such that ; then Note that, in particular, if is an open subset of such that then so that (although there are other ways in which may equal 0). Indeed,

The essential range of is defined to be the complement of It is the smallest closed subset of that contains for almost all (that is, for all except for those in some set such that ).[3] The essential range is a closed subset of so that if it is also a bounded subset of then it is compact.

The function is essentially bounded if its essential range is bounded, in which case define its essential supremum, denoted by to be the supremum of all as ranges over the essential range of [3]

Space of essentially bounded functions

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Let be the vector space of all bounded complex-valued -measurable functions which becomes a Banach algebra when normed by The function is a seminorm on but not necessarily a norm. The kernel of this seminorm, is a vector subspace of that is a closed two-sided ideal of the Banach algebra [3] Hence the quotient of by is also a Banach algebra, denoted by where the norm of any element is equal to (since if then ) and this norm makes into a Banach algebra. The spectrum of in is the essential range of [3] This article will follow the usual practice of writing rather than to represent elements of

Theorem[3] — Let be a resolution of identity on There exists a closed normal subalgebra of and an isometric *-isomorphism satisfying the following properties:

  1. for all and which justifies the notation ;
  2. for all and ;
  3. an operator commutes with every element of if and only if it commutes with every element of
  4. if is a simple function equal to where is a partition of and the are complex numbers, then (here is the characteristic function);
  5. if is the limit (in the norm of ) of a sequence of simple functions in then converges to in and ;
  6. for every

Spectral theorem

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The maximal ideal space of a Banach algebra is the set of all complex homomorphisms which we'll denote by For every in the Gelfand transform of is the map defined by is given the weakest topology making every continuous. With this topology, is a compact Hausdorff space and every in belongs to which is the space of continuous complex-valued functions on The range of is the spectrum and that the spectral radius is equal to which is [4]

Theorem[5] — Suppose is a closed normal subalgebra of that contains the identity operator and let be the maximal ideal space of Let be the Borel subsets of For every in let denote the Gelfand transform of so that is an injective map There exists a unique resolution of identity that satisfies: the notation is used to summarize this situation. Let be the inverse of the Gelfand transform where can be canonically identified as a subspace of Let be the closure (in the norm topology of ) of the linear span of Then the following are true:

  1. is a closed subalgebra of containing
  2. There exists a (linear multiplicative) isometric *-isomorphism extending such that for all
    • Recall that the notation means that for all ;
    • Note in particular that for all
    • Explicitly, satisfies and for every (so if is real valued then is self-adjoint).
  3. If is open and nonempty (which implies that ) then
  4. A bounded linear operator commutes with every element of if and only if it commutes with every element of

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

See also

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References

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