2011/04/07 by Maurice J. Dupré, James F. Glazebrook, Dupré, Maurice J. +4
Mathematics · Physics and Astronomy · #14H70 #46L08 #53B10 #53C30 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #math-ph #math.CV #math.MP #math.OA #msc:14H70 #msc:46L08 #msc:53B10 #msc:53C30
paper · pdf · doi:10.48550/arxiv.1104.1455
arxiv created 2011/04/07 · openalex publication_date 2011/04/07 · arxiv updated 2011/04/11 · openalex created_date 2025/11/01 · openalex updated_date 2026/07/28
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.