The continuous functional calculus is one of the most important foundations of the mathematical theory of C * algebras .
In advanced theory, the applications of this functional calculus are so self-evident that they are often not even mentioned. One can say without exaggeration that the continuous functional calculus, which is also in the basic sets of Gelfand-Neumark infected, the difference between C * -algebras and general Banach, where you only a holomorphic functional calculus has accounts.
If one wants to construct a functional calculus for continuous functions on the spectrum of a Banach algebra element , it makes sense to approximate the continuous functions by polynomials according to Weierstraß's approximation theorem, to insert the element into these polynomials and to show that this approximates an element in . To approximate continuous functions on, one needs polynomials in two variables, or what amounts to the same thing, polynomials in and , where denotes the complex conjugation . If you have such a polynomial and you put in place of , it is not initially clear what should be put in place of . Because an involution of the complex numbers, we consider Banach with an involution * and sets the place of . Since the polynomial ring is commutative, one has to restrict oneself to Banach algebra elements in order to obtain a homomorphism , such elements are called normal . If there is a sequence of polynomials that converges uniformly to a continuous function, then it must still be ensured that the sequence in tends towards a limit value which one could then name. A detailed analysis of this convergence problem shows that one has to fall back on C * -algebras . These considerations lead to the so-called continuous functional calculus.
The constant functional calculus
Let be a normal element of C * -algebra with unity and let be the algebra of continuous functions on . Then there is exactly one * homomorphism with and .
is an isometric isomorphism of the sub-C * algebra generated by.
Usually one uses suggestive . Then one can prove the following:
There are formulas , for all .
Applies to everyone .
Are and , then applies .
The spectral mapping principle applies : for everyone .
So one can imagine actually inserting the Banach algebra elements into continuous functions; the obvious algebraic operations behave as expected.
The requirement for a single element is not a major limitation. If necessary, one can adjoint a unity and work in the thus enlarged C * -algebra . If then and with , then and .
Applications
The following applications are typical and very simple examples of the numerous applications of the continuous functional calculus in the theory of C * algebras:
root
Let be a normal element of a C * -algebra. Then are equivalent:
If it is positive, then the restriction of the root function to continuous, and one can form by means of functional calculus . Since only takes real values, is what follows and is evident .
If the reverse is true with self-adjoint , then where , and follow from the spectral mapping theorem .
Unitary elements
If a self-adjoint element of a C * -algebra is one element , then is unitary.
It is with , because since it is self adjoint, it follows , i.e. H. is a function on the spectrum of . It follows by means of functional calculus , i.e. H. is unitary.
swell
J. Dixmier : Les C * -algèbres et leurs représentations . Gauthier-Villars, 1969.