Gelfand duality, in its real version, establishes an equivalence between the category of compact Hausdorff spaces and the (opposite) category of real Banach R-algebras $A$ satisfying $\|a^2\|=\|a\|^2$ and such that $a^2+1$ is invertible for every $a\in A$.
In the context of continuous logic, this duality appears naturally: given a theory $T$, the set of formulas modulo $T$ carries the structure of an R-algebra of continuous functions, and the corresponding space under the duality is precisely the type space. This identification makes it possible to translate model-theoretic properties into analytic/geometric properties of the associated algebras, both for first-order and continuous theories.