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

edit

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.

Theorem

edit

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]

  •  
(linear)
  •  
(multiplicative)
  •  
(involutive)

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:

  • Existence: Since the spectrum of   in the C*-subalgebra   generated by   and   is the same as it is in  , it suffices to show the statement for  .[6] The actual construction is almost immediate from the Gelfand representation: it suffices to assume   is the C*-algebra of continuous functions on some compact space   and define  .[7]
  • Uniqueness: Since   and   are fixed,   is already uniquely defined for all polynomials  , since   is a *-homomorphism. These form a dense subalgebra of   by the Stone-Weierstrass theorem. Thus   is unique.[7]

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

edit

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

  •   for all  ;   is therefore continuous.
  •  

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]

  •   is invertible if and only if   has no zero on  .[16] Then   holds.[17]
  •   is self-adjoint if and only if   is real-valued, i.e.  .
  •   is positive ( ) if and only if  , i.e.  .
  •   is unitary if all values of   lie in the circle group, i.e.  .
  •   is a projection if   only takes on the values   and  , i.e.  .

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]

Applications

edit

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

Spectrum

edit

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

  •   is self-adjoint if and only if  .
  •   is unitary if and only if  .
  •   is a projection if and only if  .

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.  

Roots

edit

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

edit

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

edit

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

edit

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]

  • For the spectrum,   holds.
  • The projections commutate with  , i.e.  .
  • The projections are orthogonal, i.e.  .
  • The sum of the projections is the unit element, i.e.  .

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  .

Notes

edit
  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.

References

edit
  • Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. ISBN 3-540-28486-9.
  • Deitmar, Anton; Echterhoff, Siegfried (2014). Principles of Harmonic Analysis. Second Edition. Springer. ISBN 978-3-319-05791-0.
  • Dixmier, Jacques (1969). Les C*-algèbres et leurs représentations (in French). Gauthier-Villars.
  • Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
  • Kaballo, Winfried (2014). Aufbaukurs Funktionalanalysis und Operatortheorie (in German). Berlin/Heidelberg: Springer. ISBN 978-3-642-37794-5.
  • Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3.
  • Kaniuth, Eberhard (2009). A Course in Commutative Banach Algebras. Springer. ISBN 978-0-387-72475-1.
  • Schmüdgen, Konrad (2012). Unbounded Self-adjoint Operators on Hilbert Space. Springer. ISBN 978-94-007-4752-4.
  • Reed, Michael; Simon, Barry (1980). Methods of modern mathematical physics. vol. 1. Functional analysis. San Diego, CA: Academic Press. ISBN 0-12-585050-6.
  • Takesaki, Masamichi (1979). Theory of Operator Algebras I. Heidelberg/Berlin: Springer. ISBN 3-540-90391-7.
edit