The Gelfand–Naimark theorem is a fundamental result in functional analysis that connects commutative C*-algebras with Hausdorff topological spaces. It states that every commutative C*-algebra can be represented as continuous functions on a compact Hausdorff space, known as the spectrum of the algebra. This representation allows for a deeper understanding of the algebra's structure and properties. Additionally, the theorem provides a way to study C*-algebras through their maximal ideals, which correspond to points in the spectrum. This correspondence is crucial for applications in various fields, including quantum mechanics and signal processing, where C*-algebras play a significant role.