Spectral theory of normal C*-algebras

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.