Differential algebras with Banach-algebra coefficients I: From C*-algebras to the K-theory of the spectral curve

TL;DR

This paper develops a framework connecting C*-algebras, spectral curves, and K-theory, providing new tools to analyze integrable systems and their spectral invariants through operator algebra techniques.

Contribution

It introduces an operator-coefficient version of Sato's Grassmannian, linking spectral curves with K-theory via C*-algebras and KK-classes, and constructs operator-valued tau-functions.

Findings

KK classes of spectral curves are constructed

K-homology of spectral curves compared with Jacobians

Operator-valued tau-functions derived from C*-algebra extensions

Abstract

We present an operator-coefficient version of Sato's infinite-dimensional Grassmann manifold, and tau-function. In this context, the Burchnall-Chaundy ring of commuting differential operators becomes a C*-algebra, to which we apply the Brown-Douglas-Fillmore theory, and topological invariants of the spectral ring become readily available. We construct KK classes of the spectral curve of the ring and, motivated by the fact that all isospectral Burchnall-Chaundy rings make up the Jacobian of the curve, we compare the (degree-1) K-homology of the curve with that of its Jacobian. We show how the Burchnall-Chaundy C*-algebra extension of the compact operators provides a family of operator-valued tau-functions.