Integrability of Analytic Almost Complex Structures on Banach Manifolds

Beltiţă, Daniel

doi:10.1007/s10455-005-2960-z

Cited by 10 publications

(5 citation statements)

References 19 publications

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“…. , p m−1 ) ∈ A × · · · × A can be identified with U A /U {a} , and now the desired conclusion follows by Corollary 16 in [9] or Example 6.20 in [10]. (See Proposition 2.7 below for the special case of W * -algebras.)…”

Section: Proposition 22 Letmentioning

confidence: 74%

Geometric representation theory for unitary groups of operator algebras

Beltiţă¹,

Raţiu²

2007

Advances in Mathematics

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Geometric realizations for the restrictions of GNS representations to unitary groups of C * -algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach-Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.

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Section: Proposition 22 Letmentioning

confidence: 74%

Geometric representation theory for unitary groups of operator algebras

Beltiţă¹,

Raţiu²

2007

Advances in Mathematics

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“…It is well known that a finite-dimensional manifold with a complex structure J is a complex manifold, that is, a manifold modeled on with holomorphic transition maps. The same result does not hold in full generality for infinite-dimensional manifolds but in the case of real analytic Banach manifolds, the result still holds [Bel05, Theorem 7]. The Hermitian symmetric spaces we consider have a real analytic complex structure and thus are complex manifolds.…”

Section: Riemannian and Hermitian Symmetric Spaces Of Infinite Dimensionmentioning

confidence: 93%

Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

Duchesne

Lécureux²,

Pozzetti³

2021

Ergod. Th. Dynam. Sys.

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We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

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“…An example of a formally integrable complex structure on a real Banach manifold which does not admit any open subset biholomorphic to an open subset of a complex Banach manifold was recently constructed by I. Patyi in [33]. However, if M is a real analytic manifold and I a formally integrable analytic complex structure, then M can be endowed with a holomorphic atlas (see [34], and [1] for the details of this result). Note also that, in the Fréchet context, L. Lempert showed in [23] that the complex structure defined in [29] by J.E.…”

Section: Theorem 213 Let M Be a Smooth Kähler Banach Manifold Endowmentioning

confidence: 95%

Hyperkähler structures and infinite-dimensional Grassmannians

Tumpach

2007

Journal of Functional Analysis

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In this paper, we describe an example of a hyperkähler quotient of a Banach manifold by a Banach Lie group. Although the initial manifold is not diffeomorphic to a Hilbert manifold (not even to a manifold modelled on a reflexive Banach space), the quotient space obtained is a Hilbert manifold, which can be furthermore identified either with the cotangent space of a connected component Gr j res (j ∈ Z), of the restricted Grassmannian or with a natural complexification of this connected component, thus proving that these two manifolds are isomorphic hyperkähler manifolds. Moreover, Kähler potentials associated with the natural complex structure of the cotangent space of Gr j res and with the natural complex structure of the complexification of Gr j res are computed using Kostant-Souriau's theory of prequantization.

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Integrability of Analytic Almost Complex Structures on Banach Manifolds

Cited by 10 publications

References 19 publications

Geometric representation theory for unitary groups of operator algebras

Geometric representation theory for unitary groups of operator algebras

Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

Hyperkähler structures and infinite-dimensional Grassmannians

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