On the manifold of tripotents in JB∗-triples

Isidro, José M.; Stachó, László L.

doi:10.1016/j.jmaa.2004.09.009

Cited by 4 publications

(6 citation statements)

References 17 publications

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“…Conversely, let u k satisfy the properties in (9). Then u : = k u k is selfadjoint and e k u = e k ( j u j ) = e k u k .…”

Section: Proposition 32mentioning

confidence: 99%

“…The equivalence (i) ⇐⇒ (ii) ⇐⇒ (iii) is known [10]. The statement concerning U = L(H) has been established in [7] as follows: From the expression of the Peirce projectors, we have for all x ∈ U…”

Section: Proofmentioning

confidence: 99%

“…The tangent space to M at a point p is the range of the 1 2 -Peirce projector of V at p, and M admits a Riemann structure (if and) only if P 1/2 (p)V is (homeomorphic to) a Hilbert space. In [7] it has been proved that the latter occurs if and only if the rank of p or the corank of p (the rank of 1 − p, where 1 is the unit of the algebra V ) is finite. M is an ortho-complemented lattice, and the mapping p → p ⊥ : = 1 − p is an involutory homeomorphism.…”

Section: Preliminaries On Jb-algebras 1introductionmentioning

confidence: 99%

“…Later on, in [15] Nomura established similar results for the manifold of minimal projections in a topologically simple JH-algebra (a real Jordan-Hilbert algebra). The results in [1], [6] and [7] lead to the idea that the structure of a JBWalgebra V might encode information about the differential geometry of some manifolds naturally associated to it [12]. In particular, that the knowledge of the JBW-structure of V is sufficient to study the manifold of projections in V .…”

Section: Preliminaries On Jb-algebras 1introductionmentioning

confidence: 99%

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Banach manifolds of algebraic elements in the algebra $$\mathcal{L}$$ (H) of bounded linear operatorsof bounded linear operators

Isidro

2005

centr.eur.j.math.

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Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V = L(H). Using the algebraic structure of V , a torsionfree affine connection ∇ (that is invariant under the group of automorphisms of V ) is defined on every connected component M of M, which in this way becomes a symmetric holomorphic manifold that consists of projections of the same rank r, (0 ≤ r ≤ ∞). We prove that M admits a Riemann structure if and only if M consists of projections that have the same finite rank r or the same finite corank, and in that case ∇ is the Levi-Civita and the Kähler connection of M. Moreover, M turns out to be a totally geodesic Riemann manifold whose geodesics and Riemann distance are computed.

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“…Conversely, let u k satisfy the properties in (9). Then u : = k u k is selfadjoint and e k u = e k ( j u j ) = e k u k .…”

Section: Proposition 32mentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Preliminaries On Jb-algebras 1introductionmentioning

confidence: 99%

Section: Preliminaries On Jb-algebras 1introductionmentioning

confidence: 99%

See 2 more Smart Citations

Banach manifolds of algebraic elements in the algebra $$\mathcal{L}$$ (H) of bounded linear operatorsof bounded linear operators

Isidro

2005

centr.eur.j.math.

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show abstract

“…[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.…”

Section: Introductionmentioning

confidence: 99%

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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Jordan triples and Riemannian symmetric spaces

Chu

2008

Advances in Mathematics

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On the manifold of tripotents in JB∗-triples

Cited by 4 publications

References 17 publications

Banach manifolds of algebraic elements in the algebra $$\mathcal{L}$$ (H) of bounded linear operatorsof bounded linear operators

Banach manifolds of algebraic elements in the algebra $$\mathcal{L}$$ (H) of bounded linear operatorsof bounded linear operators

Grassmann manifolds of Jordan algebras

Jordan triples and Riemannian symmetric spaces

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