Gelfand transform - Imported Knowlpedia Content

Definition

Let $A$ be a commutative $C^*$ -algebra with character space $\Delta(A)$ . The Gelfand transform of $a\in A$ is the function

\widehat a:\Delta(A)\to\mathbb C,\qquad \widehat a(\chi)=\chi(a).

The assignment $\Gamma_A(a)=\widehat a$ is a $*$ -homomorphism $A\to C_0(\Delta(A))$ , also called the Gelfand representation. For $C^*$ -algebras, the commutative Gelfand--Naimark theorem states that $\Gamma_A$ is an isometric $*$ -isomorphism. Thus the transform recovers both the algebra and its norm from scalar-valued characters. Each transformed element is continuous and vanishes at infinity.

Algebraic and spectral properties

For $a,b\in A$ and $\lambda\in\mathbb C$ ,

\widehat{ab}=\widehat a\,\widehat b,\qquad \widehat{a^*}=\overline{\widehat a},\qquad \widehat{\lambda a+b}=\lambda\widehat a+\widehat b.

Moreover, $\|a\|=\|\widehat a\|_\infty$ , and the range of $\widehat a$ , with zero added when required in the nonunital case, determines $\sigma_A(a)$ . Surjectivity onto $C_0(\Delta(A))$ is the substantive $C^*$ -algebra theorem Murphy, chapter on the Gelfand transform.

Canonical model

For $A=C_0(X)$ , every character is evaluation at a unique $x\in X$ . After identifying $X$ with $\Delta(A)$ , the transform sends a function $f$ to the same function: $\widehat f(\chi_x)=f(x)$ . This model explains why multiplication, involution, and norm become pointwise multiplication, complex conjugation, and the supremum norm.

Scope

The Gelfand transform is defined more generally for commutative Banach algebras, but it need not then be isometric or surjective. For a noncommutative $C^*$ -algebra, scalar characters see only commutative quotients and can fail to separate points. The representation theorem in the core therefore requires commutativity.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: the chapter proving the Gelfand representation theorem for commutative $C^*$ -algebras.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: the introductory chapter on spectra and commutative $C^*$ -algebras.