  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.
