Polish spaces for countable and separable structures through quotient encodings

TL;DR

This paper introduces a unified Borel hierarchy framework for analyzing algebraic and analytic structures via quotient encodings, providing explicit bounds and demonstrating the method's limits.

Contribution

It develops a universal quotient-based coding scheme that bounds the Borel complexity of properties in various algebraic and analytic structures, including Banach and countable algebraic structures.

Findings

Polish kernel spaces for Banach structures with continuous quotient norms.

Explicit Borel bounds for properties like nuclearity and simplicity.

Identification of a $oldsymbol{ ext{Pi}^1_1}$-complete property outside the Borel hierarchy.

Abstract

We develop a unified framework for locating natural properties of algebraic and analytic structures within the Borel hierarchy. Objects are presented as quotients of a universal generator and definability is read directly from the quotient data. For separable Banach-type structures (Banach algebras, $C^*$-algebras, Banach lattices, TROs) the kernel space is Polish under the Wijsman topology, and the quotient-norm functional $K\mapsto \|x+K\|$ is continuous, yielding a uniform definability scheme whose Borel ranks are bounded by quantifier alternation depth. For countable algebraic structures (groups, rings, lattices) we work on compact Polish spaces of congruences where atomic predicates are clopen. We obtain explicit Borel upper bounds: in the \emph{unital} $C^*$-algebra coding based on $C^*_{\max}(F_\infty)$, stable finiteness is closed, nuclearity is Borel, simplicity is~$G_\delta$, AF-ness lies in~$\Pi^0_3$, nuclear dimension~$\le n$ lies in~$\Pi^0_3$, and for fixed exact~$D$, $D$-absorption is analytic. For countable groups, soficity is~$G_\delta$; for abelian groups, slenderness is~$\Pi^0_3$. We give an internal Borel coding of the $K_0$-assignment in the quotient/Wijsman framework; for each fixed coordinate the corresponding section is $F_\sigma$, and suspension together with Bott periodicity yields Borel codings of all higher $K$-groups. We also show that several bounds are optimal ($\Sigma^0_2$- and $\Pi^0_2$-complete). To calibrate the method's reach, we exhibit a $\Pi^1_1$-complete property (separable dual in the commutative $C^*$-setting), provably outside the Borel hierarchy.