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Differential algebras with Banach-algebra coefficients. I: From \(C^*\)-algebras to the \(K\)-theory of the spectral curve. (English) Zbl 1276.19005

Summary: We present an operator-coefficient version of Sato’s infinite-dimensional Grassmann manifold, and \(\tau\)-function. In this setting the classical Burchnall-Chaundy ring of commuting differential operators can be shown to determine a \(C^*\)-algebra. For this \(C^*\)-algebra topological invariants of the spectral ring become readily available, and further, the Brown-Douglas-Fillmore theory of extensions can be applied. 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 by the compact operators provides a family of operator \(\tau\)-functions.

MSC:

19K33 Ext and \(K\)-homology
19K35 Kasparov theory (\(KK\)-theory)
14H40 Jacobians, Prym varieties
47C15 Linear operators in \(C^*\)- or von Neumann algebras
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