Continuous functional calculus
Encyclopedia
In mathematics
, the continuous functional calculus of operator theory
and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. More precisely,
Theorem. Let x be a normal
element of a C*-algebra A with an identity element 1; then there is a unique mapping π : f → f(x) defined for f a continuous function on the spectrum Sp(x) of x such that π is a unit-preserving morphism of C*-algebras such that π(1) = 1 and π(ι) = x, where ι denotes the function z → z on Sp(x).
The proof of this fact is almost immediate from the Gelfand representation
: 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 particular, this implies that bounded self-adjoint operators on a Hilbert space
have a continuous functional calculus.
For the case of self-adjoint operators on a Hilbert space of more interest is the Borel functional calculus
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the continuous functional calculus of operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. More precisely,
Theorem. Let x be a normal
Normal operator
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operatorN:H\to Hthat commutes with its hermitian adjoint N*: N\,N^*=N^*N....
element of a C*-algebra A with an identity element 1; then there is a unique mapping π : f → f(x) defined for f a continuous function on the spectrum Sp(x) of x such that π is a unit-preserving morphism of C*-algebras such that π(1) = 1 and π(ι) = x, where ι denotes the function z → z on Sp(x).
The proof of this fact is almost immediate from the Gelfand representation
Gelfand representation
In mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
: 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
Stone-Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function...
.
In particular, this implies that bounded self-adjoint operators on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
have a continuous functional calculus.
For the case of self-adjoint operators on a Hilbert space of more interest is the Borel functional calculus
Borel functional calculus
In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus , which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2...
.