A determinant formula for Toeplitz operators associated to a minimal flow
Efton Park
TL;DR
The paper develops a determinant for the Toeplitz algebra associated to a minimal ergodic flow by leveraging the Hochs–Kaad–Schemaitat framework in a semifinite, crossed-product setting. It constructs and analyzes a subalgebra ${\mathcal T}^1(X,\alpha)$ built from $C^1(X,\alpha)$ and a semicommutator ideal, obtaining a symbol-based trace formula for commutators and a concrete determinant $d$ on ${\mathrm GL}(n,{\mathcal SC}^1(X,\alpha))$ with an explicit commutator formula. Through the algebraic $K$-theory long exact sequence, it defines a map $\Delta: K^{\mathrm{alg}}_2(C^1(X,\alpha)) \to \mathbb{R}_{>0}$, giving real-valued invariants tied to symbols via $\Delta(\{e^\phi,e^\psi\}) = \exp\big(\tfrac{1}{2\pi}\int_X \mathrm{Im}(\phi'(x)\psi(x))\,d\mu(x)\big)$. These constructions connect operator-algebraic determinants with algebraic $K$-theory in dynamics-driven Toeplitz algebras, enabling nontrivial $K$-theory obstructions to be detected through analytic determinants.
Abstract
We define a determinant on the Toeplitz algebra associated to a minimal flow, give a formula for this determinant in terms of symbols, and show that this determinant can be used to give information about the algebraic $K$-theory of functions on the underlying space.