In advanced theory, the applications of this functional calculus are so natural that they are often not even mentioned.
It is no overstatement to say that the continuous functional calculus makes the difference between C*-algebras and general Banach algebras, in which only a holomorphic functional calculus exists.
If one wants to extend the natural functional calculus for polynomials on the spectrum
on the spectrum, it seems obvious to approximate a continuous function by polynomials according to the Stone-Weierstrass theorem, to insert the element into these polynomials and to show that this sequence of elements converges to
A detailed analysis of this convergence problem shows that it is necessary to resort to C*-algebras.
These considerations lead to the so-called continuous functional calculus.
is called the continuous functional calculus of the normal element
[3] Due to the *-homomorphism property, the following calculation rules apply to all functions
:[4] One can therefore imagine actually inserting the normal elements into continuous functions; the obvious algebraic operations behave as expected.
The requirement for a unit element is not a significant restriction.
If necessary, a unit element can be adjoined, yielding the enlarged C*-algebra
of bounded operators on a Hilbert space
In the literature, the continuous functional calculus is often only proved for self-adjoint operators in this setting.
In this case, the proof does not need the Gelfand representation.
is normal and all elements of a functional calculus commutate.
[13] The spectral mapping theorem applies:
, then also with the corresponding elements of the continuous functional calculus
commutates with the continuous functional calculus.
In particular, the continuous functional calculus commutates with the Gelfand representation.
[4] With the spectral mapping theorem, functions with certain properties can be directly related to certain properties of elements of C*-algebras:[15] These are based on statements about the spectrum of certain elements, which are shown in the Applications section.
there exists a uniquely determined positive element
is defined using the continuous functional calculus, then
there is a uniquely determined self-adjoint element
defines a uniquely determined positive element
is invertible, this can also be extended to negative values of
is positive, so that the absolute value can be defined by the continuous functional calculus
, since it is continuous on the positive real numbers.
are also referred to as the positive and negative parts.
is an unitary element, with the restriction that the spectrum is a proper subset of the unit circle, i.e.
is a real-valued continuous function on the spectrum