C*-Modular Vector States

Andruchow, Esteban; Varela, Alejandro

doi:10.1007/s00020-002-1280-y

Cited by 2 publications

(1 citation statement)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We mention that the manifold structure of the quotient sets that occur in Theorem 3.4 and Proposition 3.5 could have been less explicitly described using the general results on quotients of manifolds. Applications of this alternative method in other instances can be found e.g., in [2], [4], [8], and the references therein. Now we establish the main result of this section.…”

Section: Differential Geometric Structure Of the Coadjoint Action Gro...mentioning

confidence: 99%

Poisson geometrical aspects of the Tomita-Takesaki modular theory

Beltiţă¹,

Odzijewicz²

2019

Preprint

View full text Add to dashboard Cite

We investigate some genuine Poisson geometric objects in the modular theory of an arbitrary von Neumann algebra M. Specifically, for any standard form realization (M, H, J, P), we find a canonical foliation of the Hilbert space H, whose leaves are Banach manifolds that are weakly immersed into H, thereby endowing H with a richer Banach manifold structure to be denoted by H. We also find that H has the structure of a Banach-Lie groupoid H ⇒ M + * which is isomorphic to the action groupoid U(M) * M + * ⇒ M + * defined by the natural action of the Banach-Lie groupoid of partial isometries U(M) ⇒ L(M) on the positive cone in the predual M + * , where L(M) is the projection lattice of M. There is also a presymplectic form ω ∈ Ω 2 ( H) that comes fom the scalar product of H and is multiplicative in the usual sense of finite-dimensional Lie groupoid theory. We further show that the groupoid ( H, ω) ⇒ M + * shares several other properties of finite-dimensional presymplectic groupoids and we investigate the Poisson manifold structures of its orbits as well as the leaf space the foliation defined by the degeneracy kernel of the presymplectic form ω.

show abstract

Section: Differential Geometric Structure Of the Coadjoint Action Gro...mentioning

confidence: 99%

Poisson geometrical aspects of the Tomita-Takesaki modular theory

Beltiţă¹,

Odzijewicz²

2019

Preprint

View full text Add to dashboard Cite

show abstract

Metrics in the sphere of a C*-module

Andruchow

Varela

2007

centr.eur.j.math.

Self Cite

View full text Add to dashboard Cite

Given a unital C * -algebra A and a right C * -module X over A, we consider the problem of finding short smooth curves in the sphere S X = {x ∈ X :< x, x >= 1}. Curves in S X are measured considering the Finsler metric which consists of the norm of X at each tangent space of S X . The initial value problem is solved, for the case when A is a von Neumann algebra and X is selfdual: for any element x 0 ∈ S X and any tangent vector v at x 0 , there exists a curve γ(t) = e tZ (x 0 ), Z ∈ L A (X ), Z * = −Z and Z ≤ π, such that γ(0) = x 0 andγ(0) = v, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0 , x 1 ∈ S X , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denoteb

show abstract

C*-Modular Vector States

Cited by 2 publications

References 14 publications

Poisson geometrical aspects of the Tomita-Takesaki modular theory

Poisson geometrical aspects of the Tomita-Takesaki modular theory

Metrics in the sphere of a C*-module

Contact Info

Product

Resources

About