Metric geometry in homogeneous spaces of the unitary group of a C∗-algebra: Part I—minimal curves

Durán, Carlos E.; Mata-Lorenzo, Luis E.; Recht, Lázaro

doi:10.1016/s0001-8708(03)00148-8

Cited by 36 publications

(123 citation statements)

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“…But ||Xξ|| 2 = Xξ , Xξ = −X 2 ξ , ξ = 1. Now the proof concludes with an argument already used in part I of this paper, ( [10]). It goes like this: given a curve φ(t) joining 1 to γ(L) in U (A), define ψ(t) = F (φ(t)).…”

Section: A Review Of the Metric Geometry Of The Unitary Group Of A C mentioning

confidence: 53%

“…This paper continues the work in [10], namely the study of the geometry of generalized flags (homogeneous spaces corresponding to pairs of B ⊂ A of C * -algebras). Below there is a brief description of these spaces (a more complete introduction can be found in [10]).…”

Section: Introductionmentioning

confidence: 59%

“…Below there is a brief description of these spaces (a more complete introduction can be found in [10]). The main results are theorems [II-1,II-2], which are local and global versions of Hopf-Rinow-like theorems: given points ρ 0 , ρ 1 ∈ P, there exists a minimal uniparametric group curve joining ρ 0 and ρ 1 .…”

Section: Introductionmentioning

confidence: 99%

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Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

Durán

Mata-Lorenzo

Recht

2005

Integr. equ. oper. theory

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This article focuses on the study of the metric geometry of homogeneous spaces P = U (A)/U (B) (the unitary group of a C * -algebra A modulo the unitary group of a C * -subalgebra B) where the invariant Finsler metric in P is induced by the quotient norm of A/B. Under the assumption that B is of compact type, i.e. when the unitary group is relatively compact in the strong operator topology, this work presents local and global versions of Hopf-Rinowlike theorems: given points ρ0, ρ1 ∈ P, there exists a minimal uniparametric group curve joining ρ0 and ρ1. Mathematics Subject Classification (2000). Primary 54C40, 14E20; Secondary 46E25, 20C20.

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Section: A Review Of the Metric Geometry Of The Unitary Group Of A C mentioning

confidence: 53%

Section: Introductionmentioning

confidence: 59%

See 1 more Smart Citation

Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

Durán

Mata-Lorenzo

Recht

2005

Integr. equ. oper. theory

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“…These results are related to [4] and [5], where the problem of the existence of metric geodesics is studied in the context of abstract homogeneous spaces. In those papers a quotient Finsler metric is defined.…”

Section: Introductionmentioning

confidence: 97%

Metric geodesics of isometries in a Hilbert space and the extension problem

Andruchow

Recht

Varela

2007

Proc. Amer. Math. Soc.

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Abstract. We consider the problem of finding short smooth curves of isometries in a Hilbert space H. The length of a smooth curve γ(t), t ∈ [0, 1], is measured by means of 1 0 γ(t) dt, where denotes the usual norm of operators. The initial value problem is solved: for any isometry V 0 and each tangent vector at V 0 (which is an operator of the form iXV 0 with X * = X) with norm less than or equal to π, there exist curves of the form e itZ V 0 , with initial velocity iZV 0 = iXV 0 , which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given a symmetric operatorWe also consider the problem of finding metric geodesics joining two given isometries V 0 and V 1 . It is well known that if there exists a continuous path joining V 0 and V 1 , then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining V 0 and V 1 .

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“…Given a Hermitian matrix M P M pCq we study the diagonals D M , that attain the quotient norm The matrices M`D M will be called minimal. These matrices appeared in the study of minimal length curves in the ag manifold Ppnq " U pMnpCqq {U pDnpCqq, where UpAq denotes the unitary matrices of the algebra A when Ppnq is endowed with the quotient Finsler metric of the operator norm [1]. The minimal length curves δ in Ppnq are given by the left action of U pMnpCqq on Ppnq.…”

Section: Introductionmentioning

confidence: 99%