On Grassmannians associated with JB*-triples

Kaup, Wilhelm

doi:10.1007/pl00004842

Cited by 27 publications

(42 citation statements)

References 16 publications

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“…Thus by construction ID and IM are isomorphic as manifolds, however a priori is not clear whether ID and IM are isomorphic to IP as defined by Kaup in [10], as we shall see later on. Since ID a direct submanifold in D, the topology defined on ID by the atlas X , with basis of open sets { U e,δ : e ∈ M, δ > 0} where U e,δ := { X e (u): u ∈ Z 1/2 (e), u < δ}, coincides with the topology inherited from D. Now we study the topology on IM defined by the atlas X …”

Section: Corollarymentioning

confidence: 99%

“…In that case {aL(H)a} = aL(H)a = eL(H)e with any partial isometry e such that ran(e) = ran(a) and ker(e) = ker(a). We know [10] that J w is a complemented subspace in Z if and only if w is von Neumann regular. Yet different tripotents e and f may give rise to the same principal inner ideal.…”

Section: Id(a){xyz}mentioning

confidence: 99%

“…The manifolds M and IP have been studied separately in [3] and [10]. To study their relationships we introduce the auxiliary manifold…”

Section: An Auxiliary Manifoldmentioning

confidence: 99%

“…Recall from [10] that for von Neumann regular elements a ∈ Reg(Z) the tangent space to Reg(Z) at a is T a R = Z 1 (a) ⊕ Z 1/2 (a) and a local chart is…”

Section: Theoremmentioning

confidence: 99%

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On the Manifold of Complemented Principal Inner Ideals in Jb*-Triples

Isidro¹,

Stachó²

2006

The Quarterly Journal of Mathematics

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The set IM of Neher's classes of tripotents in an arbitrary JB*-triple Z is considered and a natural complex-analytic Banach manifold structure is defined on it. The relationship between IM and the Grassmann manifold of all complemented principal inner ideals in Z is studied in detail and the smooth complete vector fields on IM are characterized as smooth complete equivariant vector fields on the manifold M of tripotents of Z.

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Section: Corollarymentioning

confidence: 99%

Section: Id(a){xyz}mentioning

confidence: 99%

“…The manifolds M and IP have been studied separately in [3] and [10]. To study their relationships we introduce the auxiliary manifold…”

Section: An Auxiliary Manifoldmentioning

confidence: 99%

“…Recall from [10] that for von Neumann regular elements a ∈ Reg(Z) the tangent space to Reg(Z) at a is T a R = Z 1 (a) ⊕ Z 1/2 (a) and a local chart is…”

Section: Theoremmentioning

confidence: 99%

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On the Manifold of Complemented Principal Inner Ideals in Jb*-Triples

Isidro¹,

Stachó²

2006

The Quarterly Journal of Mathematics

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“…For the case of certain Banach Jordan triples, this space is studied in the paper [Ka01] by W. Kaup; more precisely, Kaup considers the space of complemented principal inner ideals I in a JB * -triple (which are all Peirce 2 -spaces for a suitable idempotent, cf. loc.…”

Section: Proofmentioning

confidence: 99%

Inner ideals and intrinsic subspaces of linear pair geometries

Bertram¹,

Löwe²

2008

Advg

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Abstract. We introduce the notion of intrinsic subspaces of linear and affine pair geometries, which generalizes the one of projective subspaces of projective spaces. We prove that, when the affine pair geometry is the projective geometry of a Lie algebra introduced in [BeNe04], such intrinsic subspaces correspond to inner ideals in the associated Jordan pair, and we investigate the case of intrinsic subspaces defined by the Peirce-decomposition which is related to 5 -gradings of the projective Lie algebra. These examples, as well as the examples of general and Lagrangian flag geometries, lead to the conjecture that geometries of intrinsic subspaces tend to be themselves linear pair geometries.Contents:

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Grassmann manifolds of Jordan algebras

Chu

2006

Arch. Math.

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We show that, in a JB-algebra, the projections form a Banach manifold and also, the rank-n projections in a JBW-factor form a Riemannian symmetric space of compact type, for n ∈ N ∪ {0}.1. Introduction. The close connection between Jordan algebras and geometry is wellknown (cf. [10]). Recently, various differentiable manifolds associated with a JB*-triple have been studied in [1], [5], [6], [7], [8]. These manifolds can be regarded as infinite dimensional analogues of the Grassmann manifolds. In particular, the manifolds of finite rank projections in the algebra B(H ) of bounded operators on a Hilbert space H have been studied in [1], [5], via the complex JB*-structures of B(H ). Since these manifolds are contained in the self-adjoint part B(H ) sa of B(H ), which is a real JB-algebra, it is desirable to study them via the real structures of B(H ) sa without complexification, and moreover, to tackle the wider question of such manifolds in arbitrary JB-algebras. The object of this paper is to address these issues, and indeed, we study manifolds of projections in JB-algebras using only real Jordan algebraic structures. The merit of this alternative approach may lie in its simplicity and generality. It also unifies and clarifies some results in [1], [5]. For convenience, we regard a point as a "0-dimensional manifold".We first show that, in any JB-algebra, the projections form a real Banach manifold P, and the finite rank projections, as well as the infinite rank projections, in a JBW-algebra form submanifolds of P. In a JBW-factor A, the manifold of finite rank projections consists of a sequence of connected components:

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On Grassmannians associated with JB*-triples

Cited by 27 publications

References 16 publications

On the Manifold of Complemented Principal Inner Ideals in Jb*-Triples

On the Manifold of Complemented Principal Inner Ideals in Jb*-Triples

Inner ideals and intrinsic subspaces of linear pair geometries

Grassmann manifolds of Jordan algebras

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