Borel functional calculus
The Borel functional calculus will be constructed by extending the continuous functional calculus for arbitrary bounded Borel functions.
1 Preliminary Facts
Let us set some notation first:
will denote the spectrum (http://planetmath.org/Spectrum) of .
will denote the -algebra (http://planetmath.org/CAlgebra) of continuous functions .
will denote the -algebra of bounded Borel functions , endowed with the sup norm.
The continuous functional calculus for allows the expression to make sense for continuous functions , by the assignment of a unital *-homomorphism
that sends the identity function to . This unital *-homomorphism is in fact uniquely determined by this property (see the entry on the continuous functional calculus (http://planetmath.org/ContinuousFunctionalCalculus2) for more details).
The objective is to extend to a unital *-homomorphism .
Since is a much larger -algebra than , there is no reson to presume that there is only one extension of . Which extension would be the most natural then? It turns out that there is a unique extension that satisfies a good continuity property.
It is known that *-homomorphisms between -algebras are continuous (see this entry (http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous)), so that whenever a net converges in the sup norm to a function we will have that in the operator norm. All extensions of will automaticaly satisfy this continuity property, but this can be improved in a satisfactory manner.
Notation - Let be a compact Hausdorff space, the space of all finite regular (http://planetmath.org/OuterRegular) Borel measures in and the -algebra of all bounded Borel functions in . The weakest topology in for which integration against any measure is continuous will be reffered to as the -topology. This means that in the -topology if and only if for all .
and the -topology corresponds exactly to the weak-* topology under this identification.
We will see in the next that there is an unique extension of that is continuous from the -topology to the weak operator topology.
Just like the Stone-Weierstrass theorem (http://planetmath.org/StoneWeierstrassTheoremComplexVersion) allowed the passage from the polynomial functional calculus to the continuous functional calculus, the Riesz representation theorem (http://planetmath.org/RieszRepresentationTheoremOfLinearFunctionalsOnFunctionSpaces) will allow the passage from the latter to the Borel functional calculus.
The following result is the key for the definition of the Borel functional calculus.
Theorem 1 - Let be a normal operator in and the unital *-homomorphism corresponding to the continuous functional calculus for . Then, extends uniquely to a *-homomorphism that is continuous from the -topology to the weak operator topology. Moreover, each operator lies in strong operator (http://planetmath.org/OperatorTopologies) closure (http://planetmath.org/Closure) of the unital *-algebra generated by .
: See this attached entry (http://planetmath.org/ProofOfBorelFunctionalCalculus)
We are now able to define the Borel functional calculus:
Definition - Let be a normal operator in . Let be the unique *-homomorphism defined in Theorem 1. This *-homomorphism is denoted by
and it is called the Borel functional calculus for .
Since this functional calculus extends the polynomial functional calculus, we have that for any polynomial ,
Moreover, since lies in the strong operator closure of the unital *-algebra generated by , for any function , we see that is the strong operator limit of polynomials .
3 Borel Calculus in von Neumann Algebras
The Borel functional calculus is in fact applicable for any normal operator in any von Neumann algebra .
That is due to the fact, expressed in Theorem 1, that for every the operator belongs to the strong operator closure of the unital *-algebra generated by . Being a von Neumann algebra, is closed (http://planetmath.org/ClosedSet) in the strong operator topology, and therefore all operators belong to .
Thus, by restriction, we have in fact a *-homomorphism
satisfying the properties of Theorem 1, i.e. we have a Borel functional calculus for normal operators of a von Neumann algebra.
|Title||Borel functional calculus|
|Date of creation||2013-03-22 18:48:44|
|Last modified on||2013-03-22 18:48:44|
|Last modified by||asteroid (17536)|
|Defines||Borel functions of a normal operator|