A Banach Algebra Version of the Sato Grassmannian and Commutative Rings of Differential Operators

Abstract

We show that commutative rings of formal pseudodifferential operators can be conjugated as subrings in noncommutative Banach algebras of operators in the presence of certain eigenfunctions. Techniques involve those of the Sato Grassmannian as used in the study of the KP hierarchy as well as the geometry of an infinite dimensional Stiefel bundle with structure modeled on such Banach algebras. Generalizations of this procedure are also considered.

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

“…whereby Gr(p, A) = G(A)/G[p] (see [18,20]). Note that for p ∈ P sa , we have U as was established in [19, §5].…”

“…If we have a unitary A-module map J satisfying J 2 = 1, there is an induced eigenspace decomposition H A = H + ⊕ H − , for which H ± ∼ = H A . This leads to the (restricted) Banach algebra A = L J (H A ) as described in [20] (generalizing that of A = C in [33]). Specifically, we have The algebra A can thus be seen as a Banach *-algebra with isometric involution (when A ∼ = C we simply write L J (H)).…”

“…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 This work arises from [20,21] where we started an operator-theoretic, Banach algebra approach to the Sato-Segal-Wilson theory in the setting of Hilbert modules with the extension of the classical Baker and τ -functions to operator-valued functions. Briefly recalled, the Sato-Segal-Wilson theory [37,38] in the context of integrable Partial Differential Equations * (PDEs) produces a linearization of the time flows, as one-parameter groups acting on Sato's 'Universal Grassmann Manifold', via conjugation of the trivial solution (τ ≡ 0) by a formal pseudodifferential operator.…”

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

“…Recall also from [8,9] the (restricted) Banach *-algebra A = L J (H A ) (J being a unitary A-module map satisfying J 2 = 1) which henceforth we use. As in [9], we assume A to be commutative (and separable).…”

“…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 .…”