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.


If one wants to extend the natural functional calculus for polynomials on the spectrum of an element of a Banach algebra to a functional calculus for continuous functions 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 . The continuous functions on are approximated by polynomials in and , i.e. by polynomials of the form . Here, denotes the complex conjugation, which is an involution on the complex numbers.[1] To be able to insert in place of in this kind of polynomial, Banach *-algebras are considered, i.e. Banach algebras that also have an involution *, and is inserted in place of . In order to obtain a homomorphism , a restriction to normal elements, i.e. elements with , is necessary, as the polynomial ring is commutative. If is a sequence of polynomials that converges uniformly on to a continuous function , the convergence of the sequence in to an element 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.


continuous functional calculus — Let be a normal element of the C*-algebra with unit element and let be the commutative C*-algebra of continuous functions on , the spectrum of . Then there exists exactly one *-homomorphism with for and for the identity.[2]

The mapping is called the continuous functional calculus of the normal element . Usually it is suggestively set .[3]

Due to the *-homomorphism property, the following calculation rules apply to all functions and scalars :[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 . Then if and with , it follows that and .[5]

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

In functional analysis, the continuous functional calculus for a normal operator is often of interest, i.e. the case where is the 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.[8]

Further properties of the continuous functional calculus

The continuous functional calculus is an isometric isomorphism into the C*-subalgebra generated by and , that is:[7]

Since is a normal element of , the C*-subalgebra generated by and is commutative. In particular, is normal and all elements of a functional calculus commutate.[9]

The holomorphic functional calculus is extended by the continuous functional calculus in an unambiguous way.[10] Therefore, for polynomials the continuous functional calculus corresponds to the natural functional calculus for polynomials: for all with .[3]

For a sequence of functions that converges uniformly on to a function , converges to .[11] For a power series , which converges absolutely uniformly on , therefore holds.[12]

If and , then holds for their composition.[5] If are two normal elements with and is the inverse function of on both and , then , since .[13]

The spectral mapping theorem applies: for all .[7]

If holds for , then also holds for all , i.e. if commutates with , then also with the corresponding elements of the continuous functional calculus .[14]

Let be an unital *-homomorphism between C*-algebras and . Then commutates with the continuous functional calculus. The following holds: for all . 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.

In the special case that is the C*-algebra of bounded operators for a Hilbert space , eigenvectors for the eigenvalue of a normal operator are also eigenvectors for the eigenvalue of the operator . If , then also holds for all .[18]


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


Let be a C*-algebra and a normal element. Then the following applies to the spectrum :[15]

Proof.[3] The continuous functional calculus for the normal element is a *-homomorphism with and thus is self-adjoint/unitary/a projection if is also self-adjoint/unitary/a projection. Exactly then is self-adjoint if holds for all , i.e. if is real. Exactly then is unitary if holds for all , therefore . Exactly then is a projection if and only if , that is for all , i.e.


Let be a positive element of a C*-algebra . Then for every there exists a uniquely determined positive element with , i.e. a unique -th root.[19]

Proof. For each , the root function is a continuous function on . If is defined using the continuous functional calculus, then follows from the properties of the calculus. From the spectral mapping theorem follows , i.e. is positive.[19] If is another positive element with , then holds, as the root function on the positive real numbers is an inverse function to the function .[13]

If is a self-adjoint element, then at least for every odd there is a uniquely determined self-adjoint element with .[20]

Similarly, for a positive element of a C*-algebra , each defines a uniquely determined positive element of , such that holds for all . If is invertible, this can also be extended to negative values of .[19]

Absolute value

If , then the element is positive, so that the absolute value can be defined by the continuous functional calculus , since it is continuous on the positive real numbers.[21]

Let be a self-adjoint element of a C*-algebra , then there exist positive elements , such that with holds. The elements and are also referred to as the positive and negative parts.[22] In addition, holds.[23]

Proof. The functions and are continuous functions on with and . Put and . According to the spectral mapping theorem, and are positive elements for which and holds.[22] Furthermore, , such that holds.[23]

Unitary elements

If is a self-adjoint element of a C*-algebra with unit element , then is unitary, where denotes the imaginary unit. Conversely, if is an unitary element, with the restriction that the spectrum is a proper subset of the unit circle, i.e. , there exists a self-adjoint element with .[24]

Proof.[24] It is with , since is self-adjoint, it follows that , i.e. is a function on the spectrum of . Since , using the functional calculus follows, i.e. is unitary. Since for the other statement there is a , such that the function is a real-valued continuous function on the spectrum for , such that is a self-adjoint element that satisfies .

Spectral decomposition theorem

Let be an unital C*-algebra and a normal element. Let the spectrum consist of pairwise disjoint closed subsets for all , i.e. . Then there exist projections that have the following properties for all :[25]

In particular, there is a decomposition for which holds for all .

Proof.[25] Since all are closed, the characteristic functions are continuous on . Now let be defined using the continuous functional. As the are pairwise disjoint, and holds and thus the satisfy the claimed properties, as can be seen from the properties of the continuous functional equation. For the last statement, let .


  1. ^ Dixmier 1977, p. 3.
  2. ^ Dixmier 1977, pp. 12–13.
  3. ^ a b c Kadison & Ringrose 1983, p. 272.
  4. ^ a b Dixmier 1977, p. 5,13.
  5. ^ a b Dixmier 1977, p. 14.
  6. ^ Dixmier 1977, p. 11.
  7. ^ a b c d Dixmier 1977, p. 13.
  8. ^ Reed & Simon 1980, pp. 222–223.
  9. ^ Dixmier 1977, pp. 5, 13.
  10. ^ Kaniuth 2009, p. 147.
  11. ^ Blackadar 2006, p. 62.
  12. ^ Deitmar & Echterhoff 2014, p. 55.
  13. ^ a b Kadison & Ringrose 1983, p. 275.
  14. ^ Kadison & Ringrose 1983, p. 239.
  15. ^ a b Kadison & Ringrose 1983, p. 271.
  16. ^ Kaballo 2014, p. 332.
  17. ^ Schmüdgen 2012, p. 93.
  18. ^ Reed & Simon 1980, p. 222.
  19. ^ a b c Kadison & Ringrose 1983, pp. 248–249.
  20. ^ Blackadar 2006, p. 63.
  21. ^ Blackadar 2006, pp. 64–65.
  22. ^ a b Kadison & Ringrose 1983, p. 246.
  23. ^ a b Dixmier 1977, p. 15.
  24. ^ a b Kadison & Ringrose 1983, pp. 274–275.
  25. ^ a b Kaballo 2014, p. 375.