Closed equivalence relations on compact spaces and pairs of commutative C*-algebras: a Categorical Approach

Abstract

In this paper, we study a categorical extension of the classical Gelfand-Naimark duality between compact Hausdorff spaces and commutative unital C*-algebras. We establish an equivalence between the category of compact Hausdorff spaces with closed equivalence relations and the category of pairs consisting of a commutative unital C*-algebra together with one of its unital C*-subalgebras. The motivation is that Gelfand duality can be enriched by additional structure: closed equivalence relations encode quotient spaces and invariance on the topological side, while subalgebras reflect restrictions and symmetries on the algebraic side. Shilov’s theorem, which identifies closed unital self-adjoint subalgebras of C(X) with algebras of functions invariant under closed equivalence relations, provides an essential link between these settings. We introduce the category EqRel, whose objects are compact Hausdorff spaces with closed equivalence relations and whose morphisms are continuous trajectory-preserving maps, and the category C*Pairs, whose objects are pairs (A,B) with A a commutative unital C*-algebra and B ⊂ A a unital C*-subalgebra, with morphisms given by unital *-homomorphisms preserving B. Contravariant functors are defined in both directions: (X,R) → (C(X),B_R), where B_R consists of functions constant on R-classes, and (A,B) → (Σ(A),R_B), where Σ(A) is the spectrum and R_B relates characters agreeing on B. Using the Kolmogorov-Gelfand theorem, the Gelfand transform, and Shilov’s theorem, we show that these functors are mutually inverse up to morphism of functors and thus prove the categorical equivalence EqRel ≃ C*Pairsop. This result demonstrates that the geometric notion of closed equivalence relations on compact spaces is in perfect correspondence with the algebraic notion of unital subalgebras of commutative C*-algebras.