In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 operat ...
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 continuous ...
theory, a continuous functional calculus is a
functional calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral the ...
which allows the application of a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
to normal elements of a C*-algebra.
Theorem
Theorem. Let ''x'' be a
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
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 colors i ...
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 natural ...
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*-alge ...
: 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. Furthermore, the spectral mapping theorem holds:
:
Spectral mapping theorem on PlanetMath
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See also
* Borel functional calculus
In functional analysis, a branch of mathematics, the Borel functional calculus is a '' functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scop ...
* Holomorphic functional calculus
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ''f'' of a complex argument ''z'' and an operator ''T'', the aim is to construct an operator, ''f''(''T ...
References
External links
Continuous functional calculus on PlanetMath
{{DEFAULTSORT:Continuous Functional Calculus
Theory of continuous functions
C*-algebras
Functional calculus