Maurice J. Dupré

THE STIEFEL BUNDLE OF A BANACH ALGEBRA

We introduce the Stiefel bundle associated to a given Banachable algebra and study the properties of this analytic principal fiber bundle over the Grassmannian of equivalence classes of idempotents in the algebra. Our main application concerns the bounded linear operators of a Banach space. In particular, the problem of smooth parametrization of subspaces can then be reduced to one involving the smooth extension of sections.

Classifying Hilbert bundles

Abstract A Hilbert bundle ( p , B , X ) is a type of fibre space p : B → X such that each fibre p −1 ( x ) is a Hilbert space. However, p −1 ( x ) may vary in dimension as x varies in X . We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An ( m, n )-bundle over the pair ( X , A ) is a Hilbert bundle ( p , B , X ) such that the dimension of p −1 ( x ) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X ), then the problem of classifying ( m, n )-bundles over ( X , A ) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [ A , G m ( C n )]/[ X , U ( n )] ¦ A , and the isomorphism classes of ( m, n )-bundles over ( X , A ) which are trivial over X , A .

Triviality Theorems for Hilbert Modules

In a recent paper of Kasparov [K] the theory of Hilbert modules over noncommutative C* -algebras is used to establish a general theory of extensions of C*-algebras that extends results of Brown, Douglas, and Fillmore [BDF], Fillmore [F], and Pimsner, Popa, and Voiculescu [PPV]. Since the category of Hilbert C (X) -modules is equivalent to the category of Hilbert bundles over X [DD;DG], many questions of topological interest can be recast in terms of Hilbert C(X)-modules which then give rise to questions about general Hilbert modules. In particular, Kasparov’s stability theorem [K] (which plays an essential part in the proof that inverses exist in the general theory of EXT) is the noncommutative extension of a triviality theorem of Dixmier and Douady [DD, Th.4] (which itself provides the existence of classifying maps for arbitrary separable Hilbert bundles over paracompact spaces).

Holomorphic Framings for Projections in A Banach Algebra

Abstract Given a complex Banach algebra, we consider the Stiefel bundle relative to the similarity class of a fixed projection. In the holomorphic category the Stiefel bundle is a holomorphic locally trivial principal bundle over a certain Grassmann manifold. Our main application concerns the holomorphic parametrization of framings for projections. In the spatial case this amounts to a holomorphic parametrization of framings for a corresponding complex Banach space.

The nearest self‐adjoint operator

The construction of self‐adjoint matrix Re(μ) =1/2(μ+μ+) from non‐Hermitian matrix μ for the purpose of calculational in molecular quantum mechanics is discussed. (AIP)

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

We present an operator-coefficient version of Sato’s infinite-dimensional Grassmann manifold, and τ-function. In this setting the classical Burchnall–Chaundy ring of commuting differential operators can be shown to determine a C*-algebra. For this C*-algebra topological invariants of the spectral ring become readily available, and further, the Brown–Douglas–Fillmore theory of extensions can be applied. 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 by the compact operators provides a family of operator τ-functions.

Curvature of Universal Bundles of Banach Algebras

Given a Banach algebra we construct a principal bundle with connection over the similarity class of projections in the algebra and compute the curvature of the connection. The associated vector bundle and the connection are a universal bundle with attendant connection. When the algebra is the linear operators over a Hilbert module, we establish an analytic diffeomorphism between the similarity class and the space of polarizations of the Hilbert module. Likewise, the geometry of the universal bundle over the latter is studied. Instrumental is an explicit description of the transition maps in each case which leads to the construction of certain functions. These functions are in a sense pre-determinants for the universal bundles in question.

RELATIVE INVERSION AND EMBEDDINGS

Abstract Commencing from a monoidal semigroup 𝐴, we consider the geometry of the space 𝑊(𝐴) of pseudoregular elements. When 𝐴 is a Banachable algebra we show that there exist certain subspaces of 𝑊(𝐴) that can be realized as submanifolds of 𝐴. The space 𝑊(𝐴) contains certain subspaces constituting the Stiefel manifolds of framings for 𝐴. We establish several embedding results for such subspaces, where the relevant maps induce embeddings of associated Grassmann manifolds.

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

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 and then consider the classical Baker and τ-functions as operator-valued. Following from Part I we produce a pre-determinant structure for a class of τ-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 leads to an operator τ-function. We extend to this setting the operator cross-ratio which had previously been used to produce the scalar-valued τ-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes of a given projection. We link directly this cross-ratio with Fay’s trisecant identity for the τ-function. By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of an 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) provides a link to the study of the KP hierarchy.

GENERAL RELATIVITY AS AN ÆTHER THEORY

Most early twentieth century relativists — Lorentz, Einstein, Eddington, for examples — claimed that general relativity was merely a theory of the aether. We shall confirm this claim by deriving the Einstein equations using aether theory. We shall use a combination of Lorentzs and Kelvins conception of the aether. Our derivation of the Einstein equations will not use the vanishing of the covariant divergence of the stress–energy tensor, but instead equate the Ricci tensor to the sum of the usual stress–energy tensor and a stress–energy tensor for the aether, a tensor based on Kelvins aether theory. A crucial first step is generalizing the Cartan formalism of Newtonian gravity to allow spatial curvature, as conjectured by Gauss and Riemann. In essence, we shall show that the Einstein equations are a special case of Newtonian gravity coupled to a particular type of luminiferous aether. Our derivation of general relativity is simple, and it emphasizes how inevitable general relativity is, given the truth of Newtonian gravity and the Maxwell equations.