Differential Algebras with Banach-Algebra Coefficients I: from C*-Algebras to the K-Theory of the Spectral Curve

Abstract

We present an operator-coefficient version of Sato's infinite-dimensional Grassmann manifold, and τ -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 τ -functions.Mathematics Subject Classification (2010) Primary: 46L08, 19K33, 19K35. Secondary: 13N10, 14H40, 47C15

Differential Algebras with Banach-Algebra Coefficients II: The Operator Cross-Ratio Tau-Function and the Schwarzian Derivative

“…Then from the operatorvalued τ -function τ λ : Γ 1 −→ C ⊗ 1 A in §3.10 (cf. [9]), it follows that (cf. [26,24]) τ λ (t + [z]) = τ λ (t 1 + z, t 2 + z 2 /2, t 3 + z 3 /3, .…”

“…Returning to the τ -functions considered in Part I [9], we next make some corresponding observations for determinants using admissible bases in V (p, A) (cf. [2]).…”

“…where we recall the space J A (X) of monomorphisms A ⊗ A −→ B W with respect to the spectral curve X (see [9,Appendix A.4]). In keeping with some standard algebraic-geometric terminology (cf.…”

“…In [8,9] we considered a certain (complex) Banach *-algebra A modeled on the linear operators of a Hilbert module denoted H A , where A is a commutative separable C*-algebra. Letting P (A) denote the idempotents in A, in [10] we considered the geometry of the space Λ = Sim(p, A), the similarity class of p ∈ P (A), which is closely related to the Grassmanian Gr(p, A) of Part I [9] (see also [8,10]). From the transition map of a principal bundle V Λ −→ Λ, we deduced a corresponding pre-determinant denoted T .…”

“…from the analogous identity for τ W and then using(3.17).From[9, Theorem 4.5], elements of Ext(A), extensions by the Burchnall-Chaundy C*-algebra A of the compact operators, lead to the mapΥ −1 A : Ext(A) −→ Γ 1 (4.24)…”