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Advances in Mathematics 184 (2004) 342–366

Metric geometry in homogeneous spaces of theunitary group of a C�-algebra:

Part I—minimal curves

Carlos E. Duran,a Luis E. Mata-Lorenzo,b and Lazaro Rechtb,c,�

a IVIC-Matematicas, Apartado 21827, Caracas 1020A, Venezuelab Departmento de Matematicas, Universidad Simon Bolıvar, Apartado Postal 89000,

Caracas 1080-A, Venezuelac Instituto Argentino de Matematica, CONICET, Argentina

Received 13 February 2003; accepted 8 April 2003

Communicated by Gang Tian

Dedicated to Horacio Porta on his 65th birthday

Abstract

We study the metric geometry of homogeneous spaces P of the unitary group of a

C�-algebra A modulo the unitary group of a C�-subalgebra B; where the invariant Finsler

metric is induced by the quotient norm of A=B:The main results include the following. In the von Neumann algebra context, for each

tangent vector X at a point rAP; there is a geodesic gðtÞ; ’gð0Þ ¼ X ; which is obtained by the

action on r of a 1-parameter group in the unitary group of A: This geodesic is minimizing up

to length p=2:r 2003 Elsevier Inc. All rights reserved.

MSC: primary 54C40; 14E20; secondary 46E25; 20C20

Keywords: Finsler metric; Homogeneous spaces; Minimal curves; Unitary group; von Neumann algebra

ARTICLE IN PRESS

�Corresponding author. Departmento de Matematicas, Universidad Simon Bolıvar, Apartado Postal

89000, Caracas 1080-A, Venezuela. Fax: 58-212-906-3278.

E-mail addresses: [email protected] (C.E. Duran), [email protected] (L.E. Mata-Lorenzo), [email protected]

(L. Recht).

0001-8708/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0001-8708(03)00148-8

1. Introduction

1.1. The category of unitary homogeneous spaces

The aim of this work is to study the metric geometry of homogeneous spaces of the‘elliptic type’, in a C�-algebra context.

Let us elaborate on the emphasized phrase: we assume that the homogeneousspaces in question are Banach manifolds P in which the unitary group U of a C�-algebra A acts transitively, say on the left; we denote the action of gAU on rAP byLgr: The isotropy Ir ¼ fgAU=Lgr ¼ rg will be required to be the unitary group of

a C�-subalgebra BCA: Thus the homogeneous spaces we study correspond to the‘relative C�-algebra category’, i.e. pairs ðA;BÞ; BCA; of C�-algebras.

The standard way of endowing a homogeneous space with a metric is to consider a(usually Riemannian) bi-invariant metric on the group and then pushing it down to thequotient P: But C�-algebras come equipped with an essentially unique norm (i.e. theoperator norm in some representation of A into the space of operators in a Hilbertspace H) which is bi-invariant although certainly non-Riemannian. This norm is quitenon-regular, in two senses: first, it is not differentiable but merely continuous. Also, it isnot strictly convex; there are many open sets of affine subspaces contained in the unitsphere, which hampers many constructions in the calculus of variations. This lack ofregularity is actually a blessing in disguise: it forces purely geometric constructions(i.e. metric geometry a la Gromov), instead of playing around with tensors anddifferential equations which sometimes obscure the underlying geometry.

There remains the question of how to push down the metric to P: The natural way

to do it is to consider each tangent space ðTPÞr as the Banach quotient ðTPÞr ¼ðTUÞ1=ðTIrÞ1 ¼ Aant=Bant; where Aant and Bant denote the antisymmetric parts of

the algebras A and B; respectively. So we define the Finsler norm in P by jjX jj ¼infbABant jZ þ bj; where Z projects to X in the quotient. We denote by j j the norm inthe C�-algebra A:

In this paper we will mainly endow ðTPÞr with the quotient metric, but the results have

easy partial generalizations to metrics that are topologically equivalent to the quotientmetric (an important instance of which are ‘connection metrics’, defined in the next section).

1.2. Generalized flags

In the finite dimensional, multiplicity-free case, assuming that A ¼ MnðCÞ; theinclusion of the unitary group of the C�-subalgebra B can be presented as a diagonalinclusion in UðnÞ of the form

Uðk1ÞUðk2Þ

&

UðkmÞ

0BBB@1CCCA

as it is well known. The homogeneous space is therefore a flag manifold.

ARTICLE IN PRESSC.E. Dur !an et al. / Advances in Mathematics 184 (2004) 342–366 343

For this reason the homogeneous spaces in this paper will be called generalized

flags. For the rest this paper, we shall automatically assume that the generalized flagin question is endowed with the quotient metric in each tangent space.

A partial list of examples in the infinite dimensional case is:

* The Grassmannian of a general C�-algebra [3,6].* Finite flags of a general C�-algebra [4].* Spaces of spectral measures [2].

Each of the previous items can be thought of as a special case of a space of spectralmeasures. These examples retain some of the flavour of the flag manifolds.

All these spaces have a canonical Banach manifold structure as quotients of theunitary group of the algebra [2].

1.3. The results

The main results in this article refer to minimal curves in P; i.e. curves withminimal length joining fixed endpoints. We will call these curves geodesics. The

length of a curve rðtÞ; 0ptp1; is given by cðrÞ ¼R 1

0 jj ’rðtÞjjrðtÞ dt; where jjX jjrðtÞdenotes the Finsler norm of the tangent vector X at the point rðtÞAP:

In this article, we study the problem: given an initial velocity vector XAðTPÞr;find geodesics g satisfying ’gð0Þ ¼ X :

Let XAðTPÞr be a tangent vector to a generalized flag. We say that a vector

ZAðTUÞ1 ¼ Aant is a lift of X if ddtjt¼0LetZr ¼ X ; i.e. Z projects to X in the quotient.

Theorem I. Let P be a generalized flag. Consider rAP and XAðTPÞr: Suppose that

there exists ZAAant which is a ‘minimal’ lift of X i.e. jZj ¼ jjX jjr: Then the

uniparametric group curve gðtÞ defined by gðtÞ ¼ LetZr0 has minimal length in the class

of all curves in P joining gð0Þ to gðtÞ for each t with jtjp p2jZj:

In the case where A is a von Neumann algebra (W �-algebra), the existence of sucha minimal lift is guaranteed:

Theorem II. Let A be a W �-algebra, and let P be a generalized flag of the unitary

group of A: Let XAðTPÞr: Then there is a lift Z of X which satisfies jZj ¼ jjX jjr; and

therefore the uniparametric group curve gðtÞ ¼ LetZr has minimal length in P among

curves joining gð0Þ and gðtÞ; for each t with jtjp p2jZj:

Thus, for von Neumann algebras, for every ‘direction’ XAðTPÞr (jjX jjr ¼ 1)

there is a uniparametric group curve gðtÞ ¼ LetZr; ’gð0Þ ¼ X ; which is a minimal curveup to length p=2: Note that, due to the lack of strict convexity of the norm, theremight be other geodesics with the same initial velocity vectors.

ARTICLE IN PRESSC.E. Dur !an et al. / Advances in Mathematics 184 (2004) 342–366344

In Part II of this paper (see [9]) we study the problem of finding minimal curvesconnecting given endpoints. We prove a global Hopf–Rinow theorem:

Theorem. Let P be a generalized flag. Given points p; qAP; there exists a minimal

uniparametric group curve gðtÞ ¼ LetZ p joining p to q:

for certain types of von Neumann algebras.

1.4. Intrinsic and extrinsic geometry

The main idea in the proofs of the aforementioned theorems is the following. First,given a tangent vector XAðTPÞr; find ‘length reducing maps’ FX : P-S; where S

is the sphere of radius 1/2 of a certain Hilbert space H in which A is represented.The geodesics of S are the usual well-known great circles parametrized by arc-length, and we find that for certain lifts Z of X ; the image of gðtÞ ¼ LetZr under FX isa great circle in the sphere isometric to g: Since the maps are length reducing, itfollows that g minimizes length between its endpoints for lengths up to p=2:

The length reducing maps we shall use, factor through the Grassmannian GrðHÞ;of the C�-algebra EndðHÞ (the bounded operators of the Hilbert space HÞ in which

the given C�-algebra A is represented: P!F GrðHÞ!mx

S: Each of these maps will

reduce the length of curves, and the curves whose length is preserved are minimalcurves.

Note that these constructions are extrinsic, in principle depending on arepresentation of A into the algebra of bounded operators of a Hilbert space, andalso depending on the map F : P-GrðHÞ which is defined in terms of therepresentation. Sections 3 and 4 dwell upon this extrinsic metric results. Then, inSections 5 and 6 we use the GNS theory of representations of C�-algebras toconstruct representations based on intrinsic data.

A simple particular case of this idea was used in [15].

1.5. Some precedents

The metric properties of homogeneous spaces of groups of invertible elements in aC�-algebra, have been useful in operator theory:

* Segal’s inequality turns out to be equivalent to a metric property (expansivity) ofthe exponential mapping of the canonical connection in the homogeneous spaceof positive invertible elements in a C�-algebra (see [5]).

* Also in the homogeneous space of positive invertible elements, for geodesics g andd; the Finsler distance function between gðtÞ and dðtÞ turns out to be a convexfunction of the parameter t: This fact leads to interesting operator inequalities(see [7]).

* Given a C�-algebra A, a C�-subalgebra B of A and a conditional expectationF : A-B; it is shown in [16] the following ‘relative polar decompositiontheorem’: Every invertible element gAA can be uniquely expressed in the form


g ¼ bcu where u is unitary, b is a positive invertible element in B and c is theexponential of a symmetric element in the kernel of F:

We remark that these examples are of ‘hyperbolic nature’ i.e. the spaces involvedhave ‘non-positive curvature’. In this paper the authors endeavour to begin the studyof the elliptic case.

2. Preliminaries and notation

2.1. Generalized flags

In this work a generalized flag means the following data:

* A CN Banach manifold P:* A unital C�-algebra A whose unitary group U acts transitively and smoothly on

P on the left. We indicate by Lgr; gAU; rAP; the action of U on P:* The isotropy group Ir at rAP given by, Ir ¼ fgAU j Lgr ¼ rg; is the unitary

group Ur of a C�-subalgebra Br of A:* The derivative ðTPrÞ1 : ðTUÞ1 ¼ Aant-ðTPÞr of the natural mapping

Pr : U-P given by PrðgÞ ¼ LgðrÞ is surjective (hence open by the Banach

Open Mapping Theorem).* The Finsler structure in P is given by jjX jjr ¼ inffjZ þ bj: bABant

r g; where

ðTPrÞ1ðZÞ ¼ X ; i.e. for rAP; the norm jjX jjr is the Banach quotient norm of X

in ðTUÞ1=ðTIrÞ1 ¼ Aant=Bantr : Observe that this Finsler structure is invariant

under the action of U:

Definition 2.1. A curve g : I-P of the form gðtÞ ¼ LetZr for ZAAant and tAI ¼½a; b�CR is called a uniparametric group curve.

Definition 2.2. We say that ZAAant is a lift of XAðTPÞr; if ðTPrÞ1ðZÞ ¼ X :

Observe that if Z is a lift of XAðTPÞr; then the uniparametric group curve

gðtÞ ¼ LetZr satisfies gð0Þ ¼ r and ’gð0Þ ¼ X :

2.2. Metric structures on generalized flags

Let M be a Banach manifold. A Finsler structure on M is a continuous selection ofnorms jj:jjm on each tangent space TMm:

Remark. The usual definition of a Finsler structure includes differentiability andstrict convexity of the norm; this notion is too restrictive in the cases we are dealing

with. See Section 9 of [10] for a discussion of C0 calculus of variations. Let us remarkthat the lack of differentiability prompts us to apply direct metric (as opposed to


topological) methods. We will call regular Finsler structures those which aredifferentiable and strictly convex.

Recall that for any Finsler structure, the length of a curve wðtÞ defined for aptpb

is given by,

cðwÞ ¼Z b

a

jj ’wðtÞjjwðtÞ dt:

The distance d in M is given as follows: we let Rr0;r1be the set of piecewise smooth

paths w (w : ½0; 1�-M) which join wð0Þ ¼ r0 to wð1Þ ¼ r1: We set,

dðr0; r1Þ ¼ inffcðwÞ j wARr0;r1g:

Definition 2.3. We say that a curve w is minimal in M if its length is the distancebetween its endpoints.

Given a Finsler metric on a space E and a submersion p : E-B (e.g. the projectionto a quotient), the most natural metric on B is this quotient metric. These metricshave been considered in the finite-dimensional regular case in [1]. There, thegeodesics on the base are projections of certain ‘horizontal’ geodesics in E:

Another example of an invariant Finsler structure arises by using an invariantconnection (reductive structure) on the bundle U-P to push down the operatornorm on U: Consider P to be a homogeneous reductive space (see [13]) over U: Wecall K the 1-form (reductive structure) in P: For each rAP and XAðTPÞr we have

KrðX ÞAAant: The reductive Finsler structure on P is defined at each r by the

reductive norm, jj jjK ; given as follows: for XAðTPÞr;

jjX jjK ¼ jKrðXÞj:

Reductive structures arise in the C�-algebra context in the presence of a conditional

expectation in the algebra, and thus their study can be considered as the geometryassociated to conditional expectations (see for example [16]).

It is clear that, for any reductive structure K ; this norm satisfies jjX jjKXjjX jj:We will present some results concerning norms other than the quotient norm inSection 8.

2.3. Grassmann representations and isotropic reflections

We are interested in equivariant maps of P into Grassmann manifolds. This willbe accomplished with the use of adapted representations, similar in spirit to the‘class 1 representations’ [11], which give equivariant embeddings of homogeneousspaces into spheres.

Definition 2.4. Let r0AP be fixed and let r0 be a unitary reflection of a Hilbert space

H; i.e. r�0 ¼ r0 : H-H and r20 ¼ 1: We say that a representation of A into EndðHÞ


is a Grassmann representation at r0 with respect to r0; if for each element gAUr0

its image (or representative) eggAEndðHÞ is in the commutant of r0 inEndðHÞ; i.e. eggr0 ¼ r0egg: We shall call this reflection r0 an isotropic reflection adaptedto r0:

Remark. In this work a representation of a C�-algebra in a Hilbert space may not befaithful.

Let GrðHÞ denote the Grassmann manifold of H which is just the set of unitaryreflections of H (as in [6]). The Grassmannian GrðHÞ is a homogeneous space underthe unitary group of the C�-algebra EndðHÞ: By means of the representationassumed above, we can think GrðHÞ as a homogeneous space under the unitarygroup U of A: For any Grassmann representation at r0 with respect to r0; weconsider the mapping F : P-GrðHÞ; given as follows: take any gAU which satisfiesr ¼ Lgr0; then

FðrÞ ¼ eggr0egg 1:

F is well defined because of the hypothesis about the commutant of r0: In fact,suppose that giAUr0

for i ¼ 1; 2 satisfy r ¼ Lgir0: We must check that

*g1r0 *g1 1 ¼ *g2r0 *g2

1

and this is equivalent to the identity,

*g2 1 *g1r0 ¼ r0 *g2

1 *g1:

But this is immediate from the assumption about the commutant of r0; for

g 12 g1AUr0

; and *g2 1 *g1 ¼ gg 1

2 g1g 12 g1:

Lemma 2.1. Each mapping F is compatible with the action of U on the corresponding

homogeneous spaces, i.e. for all gAU and rAP; FðLgrÞ ¼ eggFðrÞegg 1:

Proof. In fact, if FðLgrÞ ¼ *g1r0 *g1 1; where g1AU satisfies Lgr ¼ Lg1

r0; then r ¼Lg 1g1

r0 and FðrÞ ¼ gg 1g1g 1g1r0ð gg 1g1g 1g1Þ 1: Then eggFðrÞegg 1 ¼ *g1r0 *g1 1 ¼ FðLgrÞ; as

claimed. &

2.4. Some examples

Let us give a couple of examples which illustrate the spirit of the construction.Full flags: Let F be the homogenous space of full flags in Cn; that is, F is the set of

all p ¼ ðp1; p2;y; pnÞ where the piAEndðCnÞ are mutually orthogonal, rank 1projections (so that p1 þ p2 þ?þ pn ¼ 1). The unitary group UðnÞCEndðCnÞ actsnaturally and transitively on F:


Equivariant mappings s from F into Grassmann manifolds can be obtained asfollows: given a subset S of f1;y; ng; we define the function sS from F into aGrassmann manifold by

sSðp1;y; pnÞ ¼1

21 þ

XjAS

pj

!:

In this example the representation of EndðCnÞ is the identity. Given p0AF its imager0 ¼ sSðp0Þ is an isotropic reflection adapted to p0:

Spectral measures: The previous example may be generalized as follows. Consider theset M of spectral measures m defined on an algebra of sets in a measure space O; withvalues in the set of projections of a C�-algebra A: The natural action of the unitary

group U of A is given by ðLgmÞðMÞ ¼ gmðMÞg 1 for M measurable in O (see [2]).

Orbits of this action are homogenous spaces of the type we are considering. Fixing anelement M of the measurable algebra of O; we have a map QM : M-GrðAÞ; given byQMðmÞ ¼ mðMÞ: Note that in the previous example O ¼ f1;y; ng:

3. Length reducing maps

We present sufficient conditions to construct length reducing maps fromgeneralized flag s to Grassmann manifolds of the form GrðHÞ ¼ GrðEndðHÞÞwhere H is some Hilbert space.

Recall that the Finsler structure of the homogeneous space GrðHÞ is obtained asfollows. For XAðT GrðHÞÞr we have,

jjX jjr ¼ 12jX j;

where X is identified with an element of EndðHÞ (see [6]).

Proposition 3.1. Let P be a generalized flag. For any Grassmann representation of Aat r0 with respect to r0; the corresponding mapping F reduces length,i.e. jjðTFÞrðX ÞjjFðrÞpjjX jj; for all rAP; XAðTPÞr:

Proof. By the equivariance of F with respect to the (isometric) actions of the unitarygroup (see Lemma 2.1), and the invariance of the Finsler structure, it is enough toverify the result in r ¼ r0: Let XAðTPÞr0

: Consider any curve gðtÞ by r0 which

verifies ’gð0Þ ¼ X ; for example we consider gðtÞ ¼ LetZr0; where Z is some lift of X :This curve gðtÞ is transformed into

FðgðtÞÞ ¼ fetZetZFðr0Þge tZe tZ ¼ eteZZr0e teZZ;hence,

ðTFÞr0ðX Þ ¼ d

dt

��t¼0

FðgðtÞÞ ¼ ½ eZZ; r0�:


Consider the orthogonal decomposition induced by r0; H ¼ Hþ"H ; with Hþ ¼fx j r0ðxÞ ¼ xg and H ¼ fx j r0ðxÞ ¼ xg: With respect to this decomposition, wewrite

r0 ¼1 0

0 1

!; eZZ ¼

a11 a12

a�12 a22

!;

where a11 and a22 are antisymmetric maps a11 : Hþ-Hþ and a22 : H -H respectively. Then we have,

ðTFÞr0ðXÞ ¼ eZZ; r0

h i¼ 2

0 a12

a�12 0

!:

Since jjX jjr0¼ inffjZj: Z is a lift of Xg; for any e40; we can choose Z so that

jZjojjX jjr0þ e: Now we have, jjX jjr0

þ e4jZjXjj eZZjj and jjðTFÞr0ðXÞjjr0

¼12jj½ eZZ; r0�jj: To complete the proof, we just have to use the Lemma 3.2 below, which

asserts that:

a11 a12

a�12 a22

!��

��X 0 a12

a�12 0

!��

��;

for then we get jjX jjr0þ e4jjðTFÞr0

ðXÞjjr0which implies jjX jjr0

XjjðTFÞr0ðX Þjjr0

because e40 can be chosen arbitrarily small. &

Lemma 3.2. Let H ¼ S0 þ S1 be an orthogonal decomposition of a Hilbert space H:Consider bounded operators X and Y on H which, with respect to this orthogonal

decomposition, are represented by matrices of operators as

X ¼a b

b� c

!and Y ¼

0 b

b� 0

!:

Then jjX jjXjjbjj ¼ jjY jj; where jj jj indicates the usual norm for operators.

Proof. Recall jjY jj2 ¼ jjYY �jj; but

jjYY �jj ¼bb� 0

0 b�b

!��

�� ¼ jjbb�jj ¼ jjbjj2:

Hence, jjY jj ¼ jjbjj: Now we show that jjX jjXjjbjj: For any xAH0 with jjxjj ¼ 1;

jjb�xjjpax

b�x

!��

�� ¼ a b

b� c

!x

0

!��

��:

This proves the lemma. &


Next, we present length reducing maps mx : GrðHÞ-S from GrðHÞ into the

unit sphere S ¼ fZAH j jjZjj ¼ 1g: We consider S as a Riemannian manifold with

the metric given by jjW jjZ ¼ 12jjW jj; for each ZAS and WATSZ:

For each xAS; consider the evaluation mapping mx : GrðHÞ-S given by:

mxðrÞ ¼ rðxÞ:

Proposition 3.3. The mapping mx : GrðHÞ-S reduces length.

Proof. Consider a curve rðtÞ in GrðHÞ with ’rð0Þ ¼ YATðGrðHÞÞrð0Þ: The image

curve is mxðrðtÞÞ ¼ rðtÞx; then

ðTmxÞrð0ÞðY Þ ¼ Yx:

Then

jjðTmxÞrð0ÞðYÞjjrð0Þx ¼ jjYxjjrð0Þx

¼ 12jjYxjj

Now it follows that

jjðTmxÞrð0ÞðY Þjjrð0Þxp12jjY jj ¼ jjY jjrð0Þ: &

For xAS; we define the map Fx; Fx : P-S by

FxðrÞ ¼ FðrÞx:

Observing that Fx is the composition: P!F GrðHÞ!mx

S we get,

Corollary 3.4. Let P be a generalized flag. For any Grassmann representation of A at

r0 with respect to r0; and any xAH; the mapping Fx reduces length.

4. Geometric conditions for minimality

In this section we give ‘extrinsic’ conditions which guarantee that a short enougharc of a uniparametric group curve gðtÞ ¼ LetZr0 minimizes length among all curveswith the same end-points. The word ‘extrinsic’ here means that these conditions referto Grassmann representations.

These conditions are presented in two ways in Theorems 4.1 and 4.3.

4.1. Minimality conditions for a given Grassmann representation

Let P be a generalized flag together with a Grassmann representation at r0 withrespect to r0: Let XAðTPÞr0

and a lift Z of X be given.


Definition 4.1. The pair ðX ;ZÞ is in good position for the given representation, ifthere is a unit vector xAH such that the following conditions are satisfied:

1. jZj ¼ jjX jjr0:

2. x norms Z2; i.e. fZ2Z2x ¼ l2x; l ¼ jZj:3. r0ðxÞ ¼ x; r0ð eZZxÞ ¼ eZZx

Note that, because of condition (2) above we have l ¼ jZj ¼ jj eZZjj:

Theorem 4.1. Given a Grassmann representation of P at r0 with respect to r0; and a

pair ðX ;ZÞ ðXa0Þ which is in good position for this representation, let gðtÞ ¼ LetZr0:Then, gðtÞ minimizes length between the points r0 ¼ gð0Þ and gðtÞ if

0ptpp

2jZj:

In this case,

dðgð0Þ; gðtÞÞ ¼ ct0g ¼ tjZj:

The proof is based on the following lemma:

Lemma 4.2. With the hypotheses of Theorem 4.1, Fx : P-S maps the uniparametric

group curve gðtÞ ¼ LetZr0 into a geodesic wðtÞ ¼ FxðgðtÞÞ in the sphere S:Furthermore, the length of wðtÞ from 0 to t40 coincides with the length of gðtÞ; i.e.

ct0 ¼ tjZj:

Proof. Consider m40 and ZAH with jjZjj ¼ 1 such that eZZx ¼ mZ: We have,

l2 ¼ /fZ2Z2x j xS ¼ / eZZx j eZZxS ¼ m2:

Hence l ¼ m and eZZ leaves invariant the two-dimensional subspace E generated by xand Z; as well as its orthogonal complement E>: The set B ¼ fx; Zg forms an

orthonormal base for E: The matrix M of eZZ with respect to the base B is

M ¼0 l

l 0

!:

The reflection r0 leaves stable both E and E>; and its restriction to E has matrix R0

with respect to the base B:

R0 ¼1 0

0 1

!:

Notice that the curve wðtÞ sits inside the subspace E and

etM ¼ cosðltÞI0 þ sinðltÞJ0;


where

I0 ¼1 0

0 1

!; J0 ¼

0 1

1 0

!;

and therefore wðtÞ ¼ cosð2ltÞxþ sinð2ltÞZ: So wðtÞ is a geodesic in the sphere S:Finally, we compute the length of wðsÞ from s ¼ 0 to s ¼ t;

ct0 ¼

Z t

0

jjw0ðsÞjjwðsÞ ds ¼Z t

0

1

2jjw0ðsÞjj ds

¼Z t

0

1

2jj 2l sinð2lsÞxþ 2l cosð2lsÞZjj ds

¼Z t

0

1

2ð2lÞ ds ¼ tl: &

Now we prove Theorem 4.1. Consider a curve dðsÞ in P (0pspt) which connectsr0 ¼ gð0Þ and gðtÞ; for 0ptp p

2jZj: Consider vðsÞ ¼ FxðdðsÞÞ and wðsÞ ¼ FxðgðsÞÞ:These curves join x ¼ wð0Þ to wðtÞ: We have the following inequalities:

1. ct0wpct

0v if 0ptp p2jZj; because w is a geodesic in the Riemannian manifold S:

2. ct0dXct

0v; because Fx reduces length according to Corollary 3.4.

Now we observe that, ct0g ¼ ct

0w by Lemma 4.2. Then, combining these observations

we get,

ct0g ¼ ct

0wpct0vpct

0d: &

4.2. An alternative sufficient geometric condition for minimality

Here we present a result, similar to Theorem 4.1 in the previous section, aboutuniparametric group curves which are minimal.

Definition 4.2. Let XAðTPÞr0: We say that a representation of the C�-algebra A is

adapted to X if the following conditions are satisfied:

1. There exists a lift Z of X such that jZj ¼ jjX jj:2. There is a unit vector xAH which is a norming eigenvector for fZ2Z2; i.e.

fZ2Z2x ¼ l2x with l ¼ jZj:

3. For each ebbAgUr0Ur0

the vector ebbxAH is orthogonal to eZZx:

We will say that such a Z is adapted to X :

We note that in the previous definition the representation may not be faithful.


Theorem 4.3. Given a representation of A adapted to XAðTPÞr0; the

uniparametric group curve gðtÞ ¼ LetZr0 minimizes length up to t ¼ p2jjX jj for any Z

adapted to X :

Proof. Consider a representation of A adapted to XAðTPÞr0(cf. Definition 4.2).

Define the reflection r0 in H as follows,

r0ðzÞ ¼z if zASr0

;

z if zAS>r0;

(

where Sr0is the closure of the vector space generated by the set O ¼ fzAH j z ¼ebbx; ebbAgUr0

Ur0g: Observe that the commutant of r0 contains gUr0

Ur0: In fact it is clear that the

set O is invariant under gUr0Ur0

and therefore so are Sr0and S>

r0:

Next, we observe that the pair ðX ;ZÞ is in good position with respect to r0 in the

representation (cf. Definition 4.1). In fact it enough to see that eZZxAS>r0: But forebbAgUr0

Ur0we have /ebbx j eZZxS ¼ 0; which shows that eZZx is orthogonal to O and

therefore to Sr0: Now the proof follows from Theorem 4.1. &

5. The minimality theorem

In the previous sections, we assumed a Grassmann representation of A; and theminimality theorems there depend on the equivariant map F : P-GrðHÞ given bythe representation. Now we come to one of the main results in this paper, namelyTheorem I, which gives an intrinsic condition for minimality. The proof, however, isbased on the construction of a representation and an isotropic reflection in order toapply the theorems of Section 4.

Theorem I. Let P be a generalized flag. Consider rAP and XAðTPÞr: Suppose that

there exists ZAAant which is a ‘minimal’ lift of X i.e. jZj ¼ jjX jjr: Then the

uniparametric group curve gðtÞ defined by gðtÞ ¼ LetZr0 has minimal length in the class

of all curves in P joining gð0Þ to gðtÞ for each t with jtjp p2jZj:

That is, uniparametric group curves gðtÞ ¼ LetZr0 are minimal if Z is a lift ofg0ð0Þ ¼ X which realizes the (quotient) norm of X :

Remark. In the case of W �-algebras there always exists ZAA which is a ‘minimal’lift of X (see Theorem II).

The proof of Theorem I will be presented at the end of this section after somepreliminary results.


To find a representation of A adapted to a tangent vector XAðTPÞr0; we need

states of the algebra A ‘adapted’ to X :Consider the state j : A-C of the algebra A given by the vector x

in Definition 4.2,

jðaÞ ¼ /ax j xS; 8aAA:

With this state j; we define the inner-product in A;

/a0 j aS ¼ jða�a0Þ ¼ / ea0a0x j axS:

Remark 1. The second condition (2) in Definition 4.2, namely: There is a unit vector

xAH which is a norming eigenvector for fZ2Z2; i.e.

fZ2Z2x ¼ l2x with l ¼ jZj

is equivalent to the condition:

Z2 þ l2Aker j; l ¼ jZj:

in fact, the symmetric element Z2 þ l2AA is positive because jZj2 ¼ l2: Then ð eZZ2 þl2Þx ¼ 0 is equivalent to

/ð eZZ2 þ l2Þx j xS ¼ 0:

Remark 2. The third condition (3) in Definition 4.2, namely for each ebbAgUr0Ur0

the

vector ebbxAH is orthogonal to fZxZx; i.e.

/ebbx j eZZxS ¼ 0; 8ebbAgUr0Ur0

is equivalent to the condition: For each bAUr0; ZbAker j:

Indeed to observe that,

jðZbÞ ¼ /fZbZbx j xS ¼ /ebbx j eZZxS:

Remarks 1 and 2 lead us to the following.

Definition 5.1. Let XAðTPÞr0: We say that a state j is adapted to the tangent vector

X if it admits a lift Z such that

1. jZj ¼ jjX jjr0:

2. For each bAUr0; ZbAker j:

3. Z2 þ l2Aker j; l ¼ jZj:


Clearly to have a representation of A adapted to a vector X is equivalent to theexistence of a state j adapted to the vector X :

Next, we prove a useful lemma for which we need some notation. We fix

ZAAant: Let M ¼ fbZ þ Zb0AA j b; b0ABr0g: Notice that the symmetric part

of M is

Msim ¼ fbZ Zb� j bABr0g ¼ fbZ þ Zb j bABr0

g:

Observe that condition (2) in the definition above is equivalent to require that

MsimCker j; because, as it is well known, Ur0linearly generates Br0

; and because

ker j is �-closed.

We denote by S the real subspace of Asim generated by the subset Msim and the

element Z2 þ l2 in A where l ¼ jZj:

Lemma 5.1. The following are equivalent:

1. There exists a real linear form on Asim which is a state.

2. S-C ¼ |; where C is the open cone of positive invertible elements of A:

Proof. ð2Þ-ð1Þ: By the Hann–Banach theorem, if the real vector space S does notintersect C; then there is a linear functional which is positive on C and vanishes on S:This linear functional, when normalized, is the desired state.

ð1Þ-ð2Þ: On the other hand, any state is strictly positive on invertible positive

elements of Asim; hence the cone C does not intersect S: &

The equivalence of statements (3) and (4) in Proposition 5.2 is the basis of theproof of the minimality theorem I.

Proposition 5.2. Let P be a generalized flag over the unitary group of the C�-algebra

A: Let X be a tangent vector of P and let ZAAant be a lift of X : Then with the

notation above, the following statements are equivalent:

1. There is a representation of A; adapted to X :2. There is a state j adapted to the vector X :3. S contains no invertible positive elements of A:

4. jZ2jpjZ2 þ mj; 8mAMsim:

Proof. ð1Þ3ð2Þ: Follows from Remarks 1 and 2.ð2Þ3ð3Þ: Follows from Lemma 5.1.

ð3Þ ) ð4Þ: Suppose that condition 3 is satisfied and that jZ2j4jZ2 þ m0j for some

m0AMsim: Then, thinking of the elements of A as operators, the operator l2 is larger

that Z2 þ m0; i.e. l2 þ Z2 m040 which is a positive invertible element in S andthis contradicts the hypothesis.


ð4Þ ) ð3Þ: Suppose also that there is a positive invertible element of the form

sðl2 þ Z2Þ þ m with sAR and mAMsim; then we would have

sðl2 þ Z2Þ þ mXs40; for some sAR:

Clearly we can consider s40; and then,

l2 þ Z2 þ mXt40; for some tAR:

Then l24 Z2 m as operators, and then jZ2j4jZ2 þ mj; which contradictscondition 4. &

The last tool in the proof of Theorem I is the following ‘convexity’ result.

Lemma 5.3. In the setting above, suppose that jðZ þ bÞ2jXjZ2j for all bABr0: Then

jZ2jpjZ2 þ bZ þ Zbj; 8bABr0:

This lemma has a simple geometrical interpretation which is illustrated in Fig. 1based on the fact that the expression bZ þ Zb is the derivative at t ¼ 0 of the

expression ðZ þ tbÞ2:

Proof. Consider for t40 the A valued function hðtÞ ¼ Z2 þ ððZ þ tbÞ2 Z2Þ=t:

First we show that jhðtÞjXjZ2j: Suppose on the contrary that jhðtÞjojZ2j; then the

convex combination thðtÞ þ ð1 tÞZ2 has norm jthðtÞ þ ð1 tÞZ2jojZ2j for all0oto1: Notice that

thðtÞ þ ð1 tÞZ2 ¼ tZ2 þ ðZ þ tbÞ2 Z2 þ ð1 tÞZ2

¼ðZ þ tbÞ2:

Then we get jðZ þ tbÞ2jojZ2j which contradicts the hypothesis. Observe that

limt-0 hðtÞ ¼ Z2 þ bZ þ Zb: Consider the inequality jhðtÞjXjZ2j and by taking the

limit when t-0 we get jZ2 þ bZ þ ZbjXjZ2j as desired. &

Finally we present the proof of Theorem I.

Proof of Theorem I. By Theorem 4.3, it is enough to show that there is somerepresentation of A adapted to X : By hypothesis we have jZ þ bjXjZj; for allbABr0

: Both Z and b are antisymmetric in A; and the condition jZ þ bjXjZj for all

bABr0is equivalent to jðZ þ bÞ2jXjZ2j for all bABr0

: From Lemma 5.3 we get

jZ2jpjZ2 þ bZ þ Zbj; 8bABr0: But then from Proposition 5.2 we get that there is

some representation of A adapted to X as desired. &


Remark. In Theorem I, the analytical hypothesis jðZ þ bÞjXjZj; 8bABr0implies

the geometrical condition that the pair ðX ;ZÞ is in good position in somerepresentation of the C�-algebra A:

6. Existence of minimal curves with given initial velocity

In this section we consider the question of the existence of minimal curves with agiven initial velocity vector X : From Theorem I, we need to know that there is lift Z

which realizes norm of X ; i.e. jZj ¼ jjX jjr0; or equivalently jZ þ bjXjZj for all

bABr0: The theorem below says that if A is a W �-algebra such a lift exists.

Theorem II. Let A be a W �-algebra, and let P be a generalized flag over U; the

unitary group of A: Let r0AP and consider any tangent vector XAðTPÞr0: Then:

* There is a lift Z of X which satisfies jZj ¼ jjX jjr0:

* The uniparametric group curve gðtÞ ¼ LetZr0 has minimal length in the class of all

curves in P joining gð0Þ to gðtÞ for each t with jtjp p2jZj:

The proof of Theorem II is presented at the end of this section. We shall need atheorem about minimality in the norm. Suppose that A is a W �-algebra and that Bis a weakly closed W �-subalgebra of A: Observe that the symmetric andantisymmetric parts of A and B are also weakly closed (see [17, p. 14]).

Theorem 6.1. Suppose that A is a W �-algebra and that B is a weakly closed

W �-subalgebra of A: In the quotient space Asim=Bsim; the (quotient) norm of each

class is realized by some element in that class.

ARTICLE IN PRESS

Z2 + Z b + b Z

(Z+tb)2

Z2

Z b + bZ

0

Fig. 1. The point Z2 þ bZ þ Zb lies outside the ball of radius jZ2j:

C.E. Dur !an et al. / Advances in Mathematics 184 (2004) 342–366358

Proof. Let ZAAsim and, for every natural number n; let bnABsim such that,

1. jZ þ bnj is a decreasing sequence of numbers.

2. If z ¼ inffjZ þ bj j bABsimg is the norm of the class of Z in the quotient

Asim=Bsim; then

zpjZ þ bnjpz þ 1

n: ð1Þ

It is clear that the set of bn’s is bounded for jbnjpjZ þ bnj þ jZjpz þ 1 þ jZj; hence

this set is weakly compact. Then there exists bABsim which is in the weak closure of

any tail fbk j kXng: It is clear that Z þ bABsim is in the weak closure of any tailDn ¼ fZ þ bk j kXng: The theorem will be proved once we show that jZ þ bj ¼ z:Suppose on the contrary that jZ þ bj4z; so there exists a natural number n0 suchthat jZ þ bj4z þ 1=n0: Denote by A� a predual of the W �-algebra A: Then,

jZ þ bj ¼ supfj/Z;Z þ bSj j ZAA�; jZj ¼ 1g;

where /Z;Z þ bS indicates de value of Z þ b at Z: We can choose then xAA� of unitlength, such that

jZ þ bjXj/x;Z þ bSj4z þ 1

n0:

Now Z þ b is in the weak closure of the tails Dn for any natural n: Then for any e40there exist arbitrarily large numbers n such that

jj/x;Z þ bSj j/x;Z þ bnSjjoe:

Taking e small enough, we can find some n4n0 such that j/x;Z þ bnSj is larger thatz þ 1=n0: But jZ þ bnjXj/x;Z þ bnSj; and we get that jZ þ bnj4z þ 1=n04z þ 1=n

which contradicts inequality (I) above. &

Therefore,

Corollary 6.2. Suppose that A is a W �-algebra and that B is a weakly closed

W �-subalgebra of A: In the quotient space Aant=Bant; the (quotient) norm of each

class is reached by some element in that class.

Proof of Theorem II. From Corollary 6.2 we get that there is a lift Z of X whichsatisfies jZj ¼ jjX jjr0

: The minimality of the given uniparametric group curve follows

from Theorem I. &

We conclude this section with the following observation about the diameterof any generalized flag over a von Neumann algebra as a consequence ofTheorems I and II.


Proposition 6.3. Let P be a generalized flag over a von Neumann algebra A: Then the

diameter dðPÞ of P satisfies p=2pdðPÞpp:

Proof. Let X be a non-zero tangent vector to P at any point r: Let Z be a minimallift of X : Then the distance between the points r and gð p

2jZjÞ is p=2; where gðtÞ ¼ LetZr

(by Theorem I). Therefore dðPÞXp=2: On the other hand, let r; sAP: Let gAU besuch that Lgr ¼ s: Since we are assuming that A is a von Neumann algebra, there

exists a symmetric element SAA such that eiS ¼ g and jSjpp (see [12]). Thus thecurve LeitSr has length less than or equal to p and joins r to s: &

The examples (see Section 7.4) suggests that the diameter of P ¼ UðAÞ=UðBÞ isan interesting invariant of the pair ðA;BÞ; which we plan to consider in aforthcoming paper (see also concluding remarks).

7. Examples

7.1. The simplest example

The simplest possible example is given by the complex projective line CP1 ¼ S2 asa homogeneous space of Uð2Þ; and the isotropy is given by matrices of the form

eiy 0

0 eif

!:

Tangent vectors canonically lift to vectors in the Lie algebra uð2Þ of the form

0 %b

b 0

!:

The isotropy acts transitively on the directions in TxCP1; and therefore the only

invariant metrics are multiples of the canonical round metric given by gðb; bÞ ¼ jbj2:In this case, the compositions of length reducing maps collapse, and one is left with

the identity on CP1: we choose the identity of Uð2Þ as Grassmann representation,and than for any tangent vector

x ¼ ðTPÞ1

0 %b

b 0

!:

The reflection

r0 ¼1 0

0 1

!


is an isotropic reflection adapted to x: The isotropy of r0 coincides with the isotropy

of CP1; and the composition

P!F GrðHÞ!mx

S

is the identity.An important feature of this example (and in fact of all Grassmann manifolds, see

[15]) is that the quotient norm coincides with the connection norm; that is, thecanonical lift above actually realizes the minimum norm of all possible lifts.

7.2. A not so simple example

Let us consider the space P ¼ Uð3Þ=Uð1Þ � Uð1Þ � Uð1Þ of 3-flags in C3: Theisotropy is composed of diagonal unitary matrices, and a tangent vector XATrP has

a canonical lift to uð3Þ of the form

0 a b

a 0 c

b c 0

0B@1CA:

In contrast to the previous example, the canonical lift does not in general realize thequotient norm, and the problem of finding such minimal lifts is in general quite hard.There are special configurations, however, in which the canonical lift actually realizesthe norm, e.g. vectors of the form

Z ¼0 a b

a 0 0

b 0 0

0B@1CA:

For such Z and any flag rAP; the curve LetZr is minimal up to length p=2: For this

kind of vectors, we can choose as representation the identity of EndðC3Þ; and asisotropy reflection the matrix

r ¼1 0 0

0 1 0

0 0 1

0B@1CA:

With minor modifications, in this example we can substitute C3 with a generalHilbert space H; and any (finite) number of flags. More precisely, if H isdecomposed as H ¼ H1"?"Hn; any operator AAEndðHÞ can be written inmatrix form Aij with respect to the decomposition, Aij : Hj-Hi: If D is the space of

diagonals matrices, the homogeneous space UðHÞ=D is the space of n-flags in H;and the same discussion applies.


The kind of vectors for which we can actually compute the minimum on the fiber,as in Z above, can be given for any generalized flag:

7.3. Grassmann-like examples

Here we look at the case where we have a projection pAA which is in thecommutant of Br0

:

In this situation we can describe special tangent vectors as follows:

Theorem 7.1. Suppose there is a self-adjoint projector p in A which is in the

commutant of the C�-subalgebra Br0in A: Suppose that a tangent vector XAðTPÞr0

admits a lift Z such that jZj ¼ jjX jjr0; and that ZAA has degree one with respect to p;

i.e. pZ ¼ Zð1 pÞ: Then X is the initial velocity of a uniparametric group curve which

minimizes length in P up to t ¼ p2jjX jjr0

:

Proof. According to Theorem 4.3, it is enough to show that A admits arepresentation which is adapted to X : From the general theory of representationsof C�-algebras (see [14] or [12]), we can choose a representation of A in a Hilbert

space H where there is a vector Z which is norming for fZ2Z2: The orthogonal

projection p; produces in H an orthogonal decomposition in which we can write eZZin the form

eZZ ¼0 z�

z 0

!;

hence

fZ2Z2 ¼ z�z 0

0 zz�

!:

Notice that in such decomposition, Z is given as Z ¼ ðZ1Z2Þ and satisfies that,

z�zðZ1Þ ¼ l2Z1; l ¼ jZjð¼ jj eZZjjÞ:

We consider first the case Z1a0: The idea is that we can set x ¼ ðZ10

0Þ the

normalization of ðZ10Þ in H: With this x we can check that the representation

previously chosen is adapted to X : In fact, any bABr0commutes with p; and ebb is

‘‘presented’’ as a diagonal matrix of operators

ebb ¼b1 0

0 b2

!:


Hence the orthogonality condition is satisfied because

eZZx ¼0 z�

z 0

!Z010

!¼

0

zZ01

!

and

ebbx ¼b1 0

0 b2

!Z010

!¼

b1Z010

!:

In the case that Z1 ¼ 0 we proceed similarly, considering the component Z2ða0Þinstead of Z1: &

The simplest concrete case in which this situation arises non-trivially is as inSection 7.2.

A more elaborate example occurs in the context of the orbit of a spectral measure.Let m be a spectral measure, and let Mm be the orbit of m under the action of the

unitary group (cf. Section 2). The spectral measure m is a map from some measurespace O into the C�-algebra A. Let us fix an element Y0AO; and consider an

antisymmetric element aAAant: Let Z ¼ ½a; mðY0Þ�: It is easy to show that Z satisfiesthe conditions of Theorem 7.1, where the projector p is mðY0Þ: Thus the curve LetZm isa geodesic in Mm which minimizes up to length p=2:

This last example actually encompasses many others (including the one in Section7.2): just recall that the flag manifolds are also orbits of spectral measures of finitemeasure spaces.

7.4. An example concerning the diameter

It is known [15] that the diameter of Grassmann manifolds is p=2 (and thereforethese are sort of ‘‘Blaschke manifolds’’, see [8]). Also Proposition 6.3 tells us that thediameter of a generalized flag lies in the interval ½p=2; p�: The diameter of the unitarygroup, as a trivial generalized flag, is p; but it remains to study the diameter of non-trivial generalized flags. Here we show non-trivial examples of diameters close to p:

Consider the C�-algebra A ¼ Cn ¼ C�?� C; whose unitary group is the n-torus Tn: Let the subalgebra B be the scalars, and we look at the homogeneous spacePn ¼ UðAÞ=UðBÞ ¼ Tn=D where D is the diagonal of Tn:

Consider the point rAPn given by the projection of

ðe ip; e in 1

np;ye ip=n; eip=n;y; ei

n 1n

p; e ipÞ ¼ uAUðAÞ;

i.e. u ¼ diagðeikp=nÞ; k ¼ n;y; n omitting k ¼ 0: The fiber over r is given by

points of the form ur ¼ eiru ¼ diagðeiðrþkp=nÞÞ; rAR: It is easy to see that if ur ¼ eZ;Z ¼ ðiz n;yiznÞ; then at least one of the zi must be close to p if n is big enough.This shows that diamðPnÞ-p as n-N: More precisely, in [9] we show


that diamðPnÞ ¼n 1

np: It is easy to make an example along these lines on P ¼

UðAÞ=UðBÞ where A ¼Q

nAZ C; B the scalars, where the diameter is exactly p:

8. Concluding remarks

8.1. The quotient norm

The results of this paper show the relevance of the problem of effectivelycomputing quotient norms and the vectors for which the quotient norm may berealized. The reader is invited to compute the quotient norm of a generic vector inthe simplest non-trivial example to get an idea of the algebraic difficulty: let A be thespace of 3 � 3 (say Hermitian) matrices, and BCA be the subspace of diagonalmatrices (cf. Section 7).

On the other extreme of generality, let S be the reader’s favourite symmetricoperator and let mS be the spectral measure associated to S: Compute the quotientFinsler norm on the orbit of mS in terms of natural invariants of S:

8.2. Other metrics

We want to remark that Theorems I and II, which are about quotient metrics,have easy partial generalizations to other invariant metrics. Let us consider invariant(Banach) Finsler norms N on P such that, on each tangent space, are topologicallyequivalent to the quotient metric. By the Banach open mapping theorem, this isequivalent to NrðXÞXcjjX jjr; and multiplying by a positive constant if necessary

(which does not influence minimal curves), we can assume that c ¼ 1 andNrðX ÞXjjX jjr: We shall call such metrics q-equivalent metrics. An important

example of such metrics are connection metrics, defined by reductive structures (seeSection 2).

A vector XAðTPÞr will be called minimal if NrðXÞ ¼ jjX jjr:

Proposition 8.1. Theorems I and II are valid for minimal vectors in q-equivalent

metrics.

Proof. Let XAðTPÞr be a minimal vector. Let gðtÞ ¼ LetZr; where Z is a lift of X

satisfying jZj ¼ NðX Þ (¼ jjX jjr). Given t1A½0; p=2�; let sðtÞ be a curve joining gð0Þ ¼r with gðt1Þ: Then, denoting by cN the length of a curve in the Finsler structure givenby N; we have

cNðsÞXcðsÞXcðgÞ ¼ cNðgÞ ;

which shows that g is minimal in the metric N: &


These results suggest the following interesting problem: given a norm N on aquotient Q ¼ B=B0 of Banach spaces which majorizes the quotient norm,characterize the set of vectors XAQ such that NðX Þ coincides with the quotientnorm of X in Q:

An important special case of q-equivalent metrics occurs when N is the connectionnorm associated to a conditional expectation.

The problem of studying the cone of minimal lifts of vectors XAðTPÞr for a fixed

r is clearly central to our work and seems to be difficult. Partial results in thisdirection can be obtained. For example it can be shown that if ZAA is a minimal liftof X ; then both ijZj and ijZj belong to the spectrum of Z:

8.3. Operator theory and metric geometry

In the ‘hyperbolic’ case, the study of the metric geometry of C�-algebra relatedhomogeneous spaces has been fruitful, cf. Section 1.5 of the introduction. What canbe said in the ‘elliptic’ case?

In the hyperbolic case, the combinatorial topology is sort of trivial due to Cartan–Hadamard type theorems: the exponential maps are diffeomorphisms [5,7,16]. In theelliptic case, the spaces P have non-trivial ‘cut-loci’ at distances beyond p=2: What isthe significance of these sets?

8.4. Diameter of generalized flags

As we have said, the problems related to what information about the pair ðA;BÞis contained in the value of the diameter of P ¼ UðAÞ=UðBÞ is interesting.In particular,

* What is the set DC½p=2; p� of diameters of generalized flags P?* Characterize the set of generalized flags whose diameter is minimal ðp=2Þ; and

maximal ðpÞ:

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Further reading

C. Duran, L. Mata-Lorenzo, L. Recht, Natural variational problems in the Grassmann manifold of

a C�-algebra with trace, Adv. in Math. 154 (1) (2000) 196–228.


metric geometry in homogeneous spaces of the unitary group ... · metric geometry in homogeneous...

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