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Wikipedia / Continuous functional calculus

In mathematics, particularly in operator theory and C*-algebra theory, the continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra.

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.

Motivation

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 , i.e. by polynomials of the form denotes the complex conjugation, which is an involution on the in this kind of polynomial, Banach *-algebras is inserted in place of In order to obtain a homomorphism is commutative is a sequence of polynomials that converges uniformly must be ensured. 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.

Theorem

and scalars: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 and

The existence and uniqueness of the continuous functional calculus are proven separately:

  • Existence: in the C*-subalgebra, it suffices to show the statement for The actual construction is almost immediate from the Gelfand representation and define

  • Uniqueness: is a *-homomorphism. These form a dense is

In functional analysis 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

Further properties of the continuous functional calculus

is an isometric isomorphism, that is:

  • is therefore continuous.

is normal and all elements of a functional calculus

The holomorphic functional calculus is extended by the continuous functional calculus in an unambiguous for all

converges to For a power series, which converges absolutely uniformly

holds for their is the inverse function, since

The spectral mapping theorem for all

, then also with the corresponding elements of the continuous functional calculus

and for all In particular, the continuous functional calculus commutates with the Gelfand

With the spectral mapping theorem, functions with certain properties can be directly related to certain properties of elements of C*-algebras:

These are based on statements about the spectrum of certain elements, which are shown in the Applications section.

, eigenvectors of the operator also holds for all

Applications

The following applications are typical and very simple examples of the numerous applications of the continuous functional calculus:

Spectrum

a normal element. Then the following applies to the spectrum

  • is self-adjoint if and only if

  • is unitary if and only if

  • is a projection if and only if

Proof., therefore

Roots

be a positive element of a C*-algebra -th

Proof. is a continuous function on is holds, as the root function on the positive real numbers is an inverse function to the function

with

holds for all is invertible, this can also be extended to negative values of

Absolute value

, since it is continuous on the positive real

are also referred to as the

Proof. and , such that

Unitary elements

denotes the imaginary unit is a unitary element, with the restriction that the spectrum is a proper subset with

Proof. is a function on the spectrum of is a self-adjoint element that satisfies

Spectral decomposition theorem

pairwise disjoint closed, i.e. that have the following properties for all

  • holds.

  • , i.e.

  • The projections are orthogonal, i.e.

  • The sum of the projections is the unit element, i.e.

holds for all

Proof. are closed, the characteristic functions are continuous on satisfy the claimed properties, as can be seen from the properties of the continuous functional equation. For the last statement, let

Notes

References

  • English translation of