Polish spaces for countable and separable structures through quotient encodings

Abstract

We develop a quotient-based framework for locating natural properties of countable algebraic structures and separable Banach-type structures in the Borel hierarchy. The common idea is to present an object as a quotient of a fixed generator and to read definability from the corresponding kernel or congruence. For separable Banach-type structures, including Banach algebras, $C^*$-algebras and TROs, admissible kernels form Polish spaces; in the Wijsman topology the quotient-norm functional $K\mapsto \|x+K\|$ is continuous. This gives a uniform definability scheme with explicit Borel upper bounds. For countable algebraic structures, congruence spaces are compact zero-dimensional Polish spaces and atomic predicates are clopen. For Banach algebras we obtain, among other estimates, closedness of commutativity, the abstract uniform-algebra norm identity and Dedekind finiteness, and $G_δ$ bounds for topological stable-rank bounds. In the unital $C^*$-algebra coding based on $C^*_{\max}(F_\infty)$ we obtain closedness of stable finiteness and existence of a tracial state, $G_δ$ bounds for AF-ness, MF-ness, approximate divisibility and real-rank bounds; a $Π^0_3$ bound for nuclear dimension; Borelness of nuclearity and simplicity, and analyticity of $D$-absorption for fixed exact $D$. The $G_δ$ bounds for AF-ness, real-rank bounds and topological stable-rank bounds are shown to be sharp by continuous reductions from a canonical $Π^0_2$-complete set. We give an internal Borel coding of the $K_0$-assignment: every coordinate section is $F_σ$, and the resulting map into the standard subgroup coding of countable abelian groups is of Baire class~$2$. Suspension and Bott periodicity, combined with the known standard coding computations, yield Borel codings of $K_1$ and all higher $K$-groups.