# Differential algebras with Banach-algebra coefficients II: The operator
cross-ratio tau-function and the Schwarzian derivative

Research paper by **Maurice J. Dupré, James F. Glazebrook, Emma Previato**

Indexed on: **07 Apr '11**Published on: **07 Apr '11**Published in: **Mathematics - Operator Algebras**

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#### Abstract

Several features of an analytic (infinite-dimensional) Grassmannian of
(commensurable) subspaces of a Hilbert space were developed in the context of
integrable PDEs (KP hierarchy). We extended some of those features when
polarized separable Hilbert spaces are generalized to a class of polarized
Hilbert modules, in particular the Baker and tau-functions, which become
operator-valued. Following from Part I we produce a pre-determinant structure
for a class of tau-functions defined in the setting of the similarity class of
projections of a certain Banach *-algebra. This structure is explicitly derived
from the transition map of a corresponding principal bundle. The determinant of
this map gives a generalized, operator-valued tau-function that takes values in
a commutative C*-algebra. We extend to this setting the operator cross-ratio
which had been used to produce the scalar-valued tau-function, as well as the
associated notion of a Schwarzian derivative along curves inside the space of
similarity classes. We link directly this cross-ratio with Fay's trisecant
identity for the tau-function (equivalent to the KP hierarchy). By restriction
to the image of the Krichever map, we use the Schwarzian to introduce the
notion of operator-valued projective structure on a compact Riemann surface:
this allows a deformation inside the Grassmannian (as it varies its complex
structure). Lastly, we use our identification of the Jacobian of the Riemann
surface in terms of extensions of the Burchnall-Chaundy C*-algebra (Part I) to
describe the KP hierarchy.