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 H. This article describes the spectral theory of closed normal subalgebras of

Resolution of identity

Throughout, H 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 H (i.e. is a bounded linear operator that satisfies and ) such that

(where is the identity operator of H) and for every x and y in H, the function defined by is a complex measure on (that is, a complex-valued countably additive function).

A resolution of identity[1] on a measurable space is a function such that for every :

  1. ;
  2. ;
  3. for every is a self-adjoint projection on H;
  4. for every x and y in H, 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 x and y in H, 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

Throughout, let be a resolution of identity. For all x in H, is a positive measure on with total variation and that satisfies for all [1]

For every :

  • (since both are equal to ).[1]
  • If then the ranges of the maps and are orthogonal to each other and [1]
  • is finitely additive.[1]
  • If are pairwise disjoint elements of whose union is and if for all i then [1]
    • 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 0 or the partial sums cannot form a Cauchy sequence unless all but finitely many of the are 0.[1]
  • For any fixed x in H, the map defined by is a countably additive H-valued measure on
    • Here countably additive means that whenever are pairwise disjoint elements of whose union is then the partial sums converge to in H. Said more succinctly, [1]

L(π) - space of essentially bounded function

The be a resolution of identity on

Essentially bounded functions

Suppose is a complex-valued -measurable function. There exists a unique largest open subset of (ordered under subset inclusion) such that [2] 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 D 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 f is defined to be the complement of It is the smallest closed subset of that contains for almost all (i.e. for all except for those in some set such that ).[2] The essential range is a closed subset of so that if it is also a bounded subset of 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 to be the supremum of all as ranges over the essential range of f.[2]

Space of essentially bounded functions

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 [2] 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 f.[2] This article will follow the usual practice of writing f rather than to represent elements of

Theorem[2]  Let be a resolution of identity on There exists a closed normal subalgebra A of and an isometric *-isomorphism satisfying the following properties:

  1. for all x and y in H 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 f is a simple function equal to where is a partition of X and the are complex numbers, then (here is the characteristic function);
  5. if f is the limit (in the norm of ) of a sequence of simple functions in then converges to in and ;
  6. for every

Spectral theorem

The maximal ideal space of a Banach algebra A is the set of all complex homomorphisms which we'll denote by For every T in A, the Gelfand transform of T is the map defined by is given the weakest topology making every continuous. With this topology, is a compact Hausdorff space and every T in A, G(T) 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 [3]

Theorem[4]  Suppose A is a closed normal subalgebra of that contains the identity operator and let be the maximal ideal space of A. Let be the Borel subsets of For every T in A, let denote the Gelfand transform of T so that G is an injective map There exists a unique resolution of identity that satisfies:

for all and all ;

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 B be the closure (in the norm topology of ) of the linear span of Then the following are true:

  1. B is a closed subalgebra of containing A;
  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 f 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 A if and only if it commutes with every element of

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

See also

References

    1. Rudin 1991, pp. 316-318.
    2. Rudin 1991, pp. 318-321.
    3. Rudin 1991, p. 280.
    4. Rudin 1991, pp. 321-325.
    • Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.CS1 maint: ref=harv (link)
    • Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 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. 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.


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