Gelfand duality - Knowlpedia

Statement

For a commutative $C^*$ -algebra $A$ , let $\Delta(A)$ be its character space. Gelfand duality asserts that the Gelfand transform

A\longrightarrow C_0(\Delta(A)),\qquad a\longmapsto (\chi\mapsto\chi(a)),

is an isometric $*$ -isomorphism, while evaluation gives a homeomorphism $X\cong\Delta(C_0(X))$ for every locally compact Hausdorff space $X$ . Moreover, a proper continuous map $f:X\to Y$ corresponds contravariantly to the nondegenerate $*$ -homomorphism $f^*:C_0(Y)\to C_0(X)$ , $h\mapsto h\circ f$ .

Categorical content

The assignments $X\mapsto C_0(X)$ and $A\mapsto\Delta(A)$ define contravariant functors. The two displayed canonical maps are natural isomorphisms, so the result is an equivalence of categories, not merely a classification of objects. Every nondegenerate $*$ -homomorphism between commutative $C^*$ -algebras arises uniquely as pullback along the corresponding proper continuous map.

Compact form and examples

Restricting to compact Hausdorff spaces gives the equivalent unital form: $X\mapsto C(X)$ is dual to the category of unital commutative $C^*$ -algebras and unital $*$ -homomorphisms. For example, $\Delta(C_0(\mathbb R))$ consists exactly of point evaluations and is homeomorphic to $\mathbb R$ . More generally, the theorem recovers both the points and topology of $X$ from $C_0(X)$ .

Conventions and scope

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. Publisher record. Relevant: Theorem 2.1.10, the commutative Gelfand representation theorem.