Curvature of Universal Bundles of Banach Algebras

Dupré, Maurice J.; Glazebrook, James F.; Превиато, Эмма

doi:10.1007/978-3-0346-0158-0_10

Cited by 11 publications

(24 citation statements)

References 21 publications

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“…The following proposition summarizes certain analytic properties of Λ (which like Gr(p, A) is modeled on a complex Banach space): Proof. That Λ is an open and closed holomorphic submanifold of P (A) has been shown to be the case in [7,10](cf. [22]).…”

Section: Relationship With Projective Structures On Xmentioning

confidence: 96%

“…Here we briefly describe part of the basic geometry of [10]. Firstly, let π Λ = Π|Λ and π V = Π|V (p, A).…”

Section: The Universal Bundle With Connection Over λmentioning

confidence: 99%

“…The specific cocycle is determined by two non-vanishing sections. The underlying cocycle for the universal bundle is used to arrive at the pre-determinant structure of the τ -function which we refer to as the T-function; accordingly, its determinant Det(T) is the τ -function (see [10,18,28]). The important structure lies within the geometry of this determinant bundle, its connection and its curvature.…”

Section: Introductionmentioning

confidence: 99%

“…From the transition map of a principal bundle V Λ −→ Λ, we deduced a corresponding pre-determinant denoted T . We identify in §3 a relationship between T and T obtained via a diffeomorphism between Λ and P. In fact, as shown in [10], certain calculations involving the connection and curvature are more straightforward when performed on Λ, and that is why we choose to work mainly with predeterminants T λ and the ensuing determinants τ λ .…”

Section: Introductionmentioning

confidence: 99%

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

Dupré

Glazebrook

Превиато

2012

Complex Anal. Oper. Theory

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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 τ -functions, which become 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 gives a generalized, operator-valued τ -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 τ -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 τ -function (equivalent to the KP hierarchy). By * Partial research support under grant NSF-DMS-0808708 is very gratefully acknowledged.

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Section: Relationship With Projective Structures On Xmentioning

confidence: 96%

“…Here we briefly describe part of the basic geometry of [10]. Firstly, let π Λ = Π|Λ and π V = Π|V (p, A).…”

Section: The Universal Bundle With Connection Over λmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Dupré

Glazebrook

Превиато

2012

Complex Anal. Oper. Theory

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“…Before continuing, we note that curvatures of universal connections on infinite-dimensional principal bundles were also studied, see [17].…”

Section: Curvature and Positivity On Holomorphic Vector Bundlesmentioning

confidence: 99%

Geometric Perspectives on Reproducing Kernels

Beltiţă¹,

Galé²

2015

Operator Theory

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Reproducing Kernels and Positivity of Vector Bundles in Infinite Dimensions

Beltiţă

Galé

2016

Trends in Mathematics

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Abstract. We investigate the interaction between the existence of reproducing kernels on infinite-dimensional Hermitian vector bundles and the positivity properties of the corresponding bundles. The positivity refers to the curvature form of certain covariant derivatives associated to reproducing kernels on the vector bundles under consideration. The values of the curvature form are Hilbert space operators, and its positivity is thus understood in the usual sense from operator theory.

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Curvature of Universal Bundles of Banach Algebras

Cited by 11 publications

References 21 publications

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

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

Geometric Perspectives on Reproducing Kernels

Reproducing Kernels and Positivity of Vector Bundles in Infinite Dimensions

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