Homogeneous manifolds from noncommutative measure spaces

Andruchow, Esteban; Chiumiento, Eduardo; Larotonda, Gabriel

doi:10.1016/j.jmaa.2009.11.024

Cited by 6 publications

(11 citation statements)

References 11 publications

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“…Recall that the completeness of the metric space (Stj(v),d) was proved in [7] by different methods. Also it is worthwhile noting that a similar statement was proved in [1] for homogeneous spaces in finite von Neumann algebras with the p-norm induced by the trace. Proof.…”

Section: Remark23supporting

confidence: 63%

Metric geometry in infinite dimensional Stiefel manifolds

Chiumiento

2010

Differential Geometry and its Applications

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Let 3 be a separable Banach ideal in the space of bounded operators acting in a Hilbert space 7T and T the set of partial isometries in 7T. Fix v g T. In this paper we study metric properties of the 3-Stiefel manifold associated to v, namely Stj(v) = {v0 eZ: v-v0 e 3, j(vqV0, v*v) =0), MSC: primary 22E65 secondary 47B10, 58B20 where j(.) is the Fredholm index of a pair of projections. Let ¿Yj(7T) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in 3. Then St-j(v) coincides with the orbit of v under the action of ¿Y-jCH) z ¿Y-jCH) on T given by Keywords: Partial isometry Banach ideal Finsler metric Minimal curves (u, w) • Vo = uvow*. u, w 6 Uj(7i) and Vo 6 Stj(v). We endow 5tj(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra ofi/jCH) zT/jCH) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of ¿Yj(7T) z i/jfH) by the isotropy group. Hence this metric defines the quotient topology in Stj(v). The other results concern with minimal curves in 3-Stiefel manifolds when the ideal 3 is fixed as the compact operators in 7T. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length.

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Section: Remark23supporting

confidence: 63%

Metric geometry in infinite dimensional Stiefel manifolds

Chiumiento

2010

Differential Geometry and its Applications

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“…The proof of these facts can be found in [4,Theorem 3.2]. See also [1] for related results on the unitary group of a finite von Neumann algebras and their homogeneous spaces.…”

Section: Symmetric Normsmentioning

confidence: 91%

“…The group of unitary operators on Hilbert space carries, as any Banach-Lie group, a canonical connection without torsion ∇ X Y = 1 2 [X, Y ], whose geodesics are the one-parameter groups t → ue tz (here u is a unitary operator and z an anti-Hermitian operator). In the finite dimensional setting, the trace is available to introduce a Riemannian metric on the unitary group in a standard fashion x, y g = T r(u * x(u * y) * ) = T r(xy * ), for u * x, u * y in the Lie algebra of the group, that is, for u * x, u * y anti-Hermitian matrices.…”

Section: Introductionmentioning

confidence: 99%

Optimal Paths for Symmetric Actions in the Unitary Group

Antezana

Larotonda

Varela

2014

Commun. Math. Phys.

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Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval [a, b] ⊂ R, we study the action defined in the Lie group of n × n unitary matrices U(n) bywhere α : [a, b] → U(n) is a rectifiable curve. We prove that the one-parameter subgroups of U(n) are the optimal paths, provided the spectrum of the exponent is bounded by π. Moreover, if L is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in U(n) as well as angular metrics in the Grassmann manifold.

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“…and (G/K µ , d) is a genuine metric space. 5. If µ is a R-uniform metric, we obtain that K µ is also a closed subgroup, but now the metric dist µ is left-K µ invariant on G.…”

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confidence: 88%

Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

Larotonda

2019

Forum Mathematicum

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We study the geometry of Lie groups G with a continuous Finsler metric, assuming the existence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient M ≃ G/K. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

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Homogeneous manifolds from noncommutative measure spaces

Cited by 6 publications

References 11 publications

Metric geometry in infinite dimensional Stiefel manifolds

Metric geometry in infinite dimensional Stiefel manifolds

Optimal Paths for Symmetric Actions in the Unitary Group

Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

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