In
mathematics, particularly in
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear oper ...
and
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...
theory, a continuous functional calculus is a
functional calculus which allows the application of a
continuous function to normal elements of a C*-algebra.
Theorem
Theorem. Let ''x'' be a
normal element of a C*-algebra ''A'' with an identity element e. Let ''C'' be the C*-algebra of the bounded continuous functions on the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
σ(''x'') of ''x''. Then there exists a unique mapping π : C → A, where ''π(f)'' is denoted ''f(x)'', such that π is a unit-preserving morphism of C*-algebras and π(1) = e and π(id) = ''x'', where id denotes the function ''z'' → ''z'' on σ(''x'').
In particular, this theorem implies that bounded normal operators on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
have a continuous functional calculus.
Its proof is almost immediate from the
Gelfand representation In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things:
* a way of representing commutative Banach algebras as algebras of continuous functions;
* the fact that for commutative C*-al ...
: it suffices to assume ''A'' is the C*-algebra of continuous functions on some compact space ''X'' and define
:
Uniqueness follows from application of the
Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the si ...
. Furthermore, the spectral mapping theorem holds:
:
Spectral mapping theorem on PlanetMath
/ref>
See also
* Borel functional calculus
* Holomorphic functional calculus
References
External links
Continuous functional calculus on PlanetMath
{{DEFAULTSORT:Continuous Functional Calculus
Theory of continuous functions
C*-algebras
Functional calculus