The classification problem for operator algebraic varieties and their multiplier algebras

TL;DR

This paper investigates the complexity of classifying multiplier algebras of operator algebraic varieties, showing they are not classifiable by countable structures and establishing maximum complexity for certain cases.

Contribution

It introduces turbulence theory for Polish groupoids and applies it to demonstrate the high complexity of classifying these algebras.

Findings

Multiplier algebras are not classifiable by countable structures.

Classification problem for finite-dimensional varieties has maximum complexity.

Blaschke sequences are not smoothly classifiable up to conformal automorphisms.

Abstract

We study from the perspective of Borel complexity theory the classification problem for multiplier algebras associated with operator algebraic varieties. These algebras are precisely the multiplier algebras of irreducible complete Nevanlinna-Pick spaces. We prove that these algebras are not classifiable up to algebraic isomorphism using countable structures as invariants. In order to prove such a result, we develop the theory of turbulence for Polish groupoids, which generalizes Hjorth's turbulence theory for Polish group actions. We also prove that the classification problem for multiplier algebras associated with varieties in a finite dimensional ball up to isometric isomorphism has maximum complexity among the essentially countable classification problems. In particular, this shows that Blaschke sequences are not smoothly classifiable up to conformal equivalence via automorphisms of the disc.