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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.2 Bahía Blanca jul./dic. 2005
Infinitely many minimal curves joining arbitrarily close points in a homogeneous space of the unitary group of a C*-algebra
Esteban Andruchow, Luis E. Mata-Lorenzo, Lázaro Recht, Alberto Mendoza and Alejandro Varela
Dedicated to the memory of Ángel Rafael Larotonda (Pucho).
Abstract: We give an example of a homogeneous space of the unitary group of a C*-algebra which presents a remarkable phenomenon, in its natural Finsler metric there are infinitely many minimal curves joining arbitrarily close points.
In this paper, we give an example of a homogeneous space of the unitary group of a C-algebra which presents a remarkable phenomenon. Namely, in its natural Finsler metric there are infinitely many minimal curves joining arbitrarily close points. More precisely the homogeneous space will be called
. The unitary group
of a C
-algebra
acts transitively on the left on
. The action is denoted by
, for
and
. The isotropy
will be the unitary group of a
-subalgebra
. The Finsler norm in
is naturally defined by
, for
where
projects to
in the quotient
which is identified to the tangent space
. These definitions and notation are borrowed from [1].
This work is part of a forthcoming paper by the same authors which will contain additional results about minimal vectors. We call an element minimal vector if
In [1] the following theorem is proven.
Theorem 2.1. Let be a homogeneous space of the unitary group of a C
-algebra
. Consider
and
. Suppose that there exists
which is a minimal vector i.e.
. Then the oneparameter curve
given by
has minimal length in the class of all curves in
joining
to
for each
with
.
We will use the following notation. Let be a unital C
-algebra, and
a C
-subalgebra. Denote
and
(resp.
and
) the sets of selfadjoint (resp. anti-hermitian) elements of
and
. Let
be a Hilbert space,
the algebra of bounded operators acting on
, and
the group of invertible operators.
We call an element a minimal vector if
![∥Z ∥ ≤ ∥Z + V ∥, for all V ∈ Bah.](/img/revistas/ruma/v46n2/2a0753x.png)
Note that since for any operator, , it follows that
is minimal if and only if
![∥Z ∥ ≤ ∥Z + B ∥, for all B ∈ B.](/img/revistas/ruma/v46n2/2a0756x.png)
In view of the purpose of this paper stated in the introduction and the previous theorem, to look for minimal curves we have to find minimal vectors and therefore the following theorem is relevant.
Theorem 2.2. An element is minimal if and only if there exists a representation
of
in a Hilbert space
and a unit vector
such that
Proof. The if part is trivial. Suppose that there exist as above. Then if
,
![∥Z + B∥2 ≥ ∥ρ(Z + B) ξ∥2 = ∥ρ(Z) ξ∥2 + ∥ρ(B) ξ∥2 ≥ ∥ρ(Z) ξ∥2 = - 〈ρ(Z2)ξ,ξ〉 = ∥Z∥2.](/img/revistas/ruma/v46n2/2a0767x.png)
Suppose now that is minimal. Denote by
the closed (real) linear span of
and the operators of the form
for all possible
. Note that
is positive and
is selfadjoint, i.e.
.
Denote by the cone of positive and invertible elements of
. We claim that the minimality condition implies that
. Indeed, otherwise, since
is open, there would exist
and
such that
![s(Z2 + ∥Z ∥2I) + ZB - B *Z ≥ rI, with r > 0.](/img/revistas/ruma/v46n2/2a0782x.png)
We may suppose that , so that dividing by
we get that for given
,
,
(2.1)
Also note that , then for
,
![n(Z2 + ∥Z ∥2) + ZB - B *Z ≥ Z2 + ∥Z ∥2I + ZB - B *Z ≥ rI.](/img/revistas/ruma/v46n2/2a0790x.png)
Or equivalently, dividing by ,
![( ) ( ) 2 2 1- 1- * ′ Z + ∥Z ∥ I + Z n B - nB Z ≥ r I.](/img/revistas/ruma/v46n2/2a0792x.png)
In other words, one can find with arbitrarily small norm such that inequality 2.1 holds.
This inequality clearly implies that
![sp(Z2 + ZB - B *Z) ⊂ (- ∥Z ∥2,+ ∞).](/img/revistas/ruma/v46n2/2a0794x.png)
On the other hand, since can be chosen with arbitrarily small norm, and
is non positive, it is clear that one can choose
in order that
. Therefore there exists
such that
. Let us show that this contradicts the minimality of
, and thus proves our claim. Indeed, this is stated in lemma 5.3 of [1]:
Lemma 2.3. If for all
, then also
.
We include its proof. Consider for ,
. Note that
. Otherwise
and then the convex combination
has norm strictly smaller than
for
. Note that
![tf(t) + (1 - t)Z2 = (Z + tB) *(Z + tB).](/img/revistas/ruma/v46n2/2a07112x.png)
That is , which contradicts the hypothesis, and the lemma is proven, as well as our claim.
We have that , with
a closed (real) linear submanifold of
and
open and convex in
. By the Hahn-Banach theorem, there exists a bounded linear functional
in
such that
![φ0(S) = 0 and φ0(C) > 0.](/img/revistas/ruma/v46n2/2a07121x.png)
The functional has a unique selfadjoint extension to
, let
be the normalization of this functional. Then clearly
is a state which vanishes on
. Let
be the GNS triple associated to this state. Note that since
,
, and therefore, by the equality part in the Cauchy-Schwartz inequality, it follows that
![2 2 ρ(Z )ξ = - ∥Z∥ ξ.](/img/revistas/ruma/v46n2/2a07130x.png)
Moreover, . Since
is selfadjoint, this means
for all
. Putting
in the place of
, one has that in fact
for all
. Then,
![0 = 〈ρ(ZB) ξ,ξ〉 = 〈ρ(B) ξ,ρ(Z)ξ〉,](/img/revistas/ruma/v46n2/2a07139x.png)
which concludes the proof.
3. Infinitely many minimal curves joining arbitrarily close points
In this example the homogeneous space is the flag manifold of 4-tuples of mutually orthogonal lines in
(1-dimensional complex subspaces). The group of unitary operators in
acts on the left in
by sending each complex line to its image by the unitary operator (thus preserving the orthogonality of the new 4-tuple complex lines). Consider the canonical flag
where
is the complex line spanned by the canonical vector
in
. The isotropy of the canonical flag
is the subgroup of 'diagonal' unitary operators.
We consider now the submanifold of
given by
![Pd = {(l1,l2,l3,l4) ∈ P ∣ sp{l1,l2}= sp{e1,e2}}](/img/revistas/ruma/v46n2/2a07152x.png)
Notice that where
is the flag manifold of couples of mutually orthogonal 1-dimensional complex lines in
. Notice also that an ordered pair of mutually orthogonal 1-dimensional complex lines in
is totally determined by the first complex line of the pair, hence
. Furthermore
, the Riemann Sphere, hence
=
.
The minimal curves presented in this example shall be constructed in . For a better geometrical view of those curves we shall identify
, via stereographic projection, with the unit sphere
in
, hence we shall make the identification
.
3.1. A description of the minimal curves. Let be the point whose coordinates are both the North Pole,
. Let
be any point of
such that
has higher latitude than
in
(
is closer to
than
).
We will fix ![]() ![]() ![]() ![]() ![]() ![]()
|
![]() |
![]() | ![]() |
3.2. A precise description of the minimal curves. To present the curves drawn above we give a more manageable description of . We consider the unitary subgroup
of the
-algebra
of
complex matrices, and denote with
the subalgebra of diagonal matrices in
. The homogeneous space
is given by the quotient
, where
is the subgroup of the diagonal unitary matrices. The group
acts on
(on the left). The tangent space at 1 (the identity class) is the subspace of anti-hermitian matrices in
with zeroes on the diagonal.
We construct as follows. First consider the subgroup
of special unitary matrices build with two,
, blocks on the diagonal. We set
as the quotient of
by the subgroup
of diagonal special unitary matrices. This submanifold is in itself a product of two copies of the quotient
of
by the subgroup of diagonal matrices in
. For the relations among the different groups here mentioned we suggest [2]. We write
and a point of
is a class (in a quotient) which in itself has two components which are also classes. We shall use the notation
.
The minimal curves starting at are of the form
where the matrices
are anti-hermitian matrices with zero trace in
built with two blocks of anti-hermitian
matrices on the diagonal (each one with zero trace).
The minimality of the curves is granted by 2.1 for the matrices shall be minimal vectors according to theorem 2.2. In fact, we shall consider
of the form
![( Z 0 ) Z = 1 0 Z2](/img/revistas/ruma/v46n2/2a07228x.png)
where and
are anti-hermitian
matrices of the form
(3.2)
(3.3)
where , and
.
The minimality of these matrices is assured in the case where
. In such case,
and, in relation to theorem 2.2, just consider the operator representation
of the
-algebra
on
, together with the unit vector
.
3.2.1. The two components of the curves in . The curve
in
has two components (in
).
We shall regard the Riemann Sphere as the complex plane
with the point "
" added. Consider a matrix
in
![( - ) a -b 2 2 u = b a- , where a, b ∈ ℂ and ∣a∣ + ∣b∣ = 1](/img/revistas/ruma/v46n2/2a07253x.png)
We consider the mapping from
to
is given by
![L(u) = a, if b ⁄= 0, else L(u) = ∞. b](/img/revistas/ruma/v46n2/2a07257x.png)
It is clear that this mapping induces an explicit diffeomorphism from the quotient of by its diagonal matrices to the Riemann Sphere
.
Consider the unit sphere in
, and let the equatorial plane,
, represent the "finite" part of the Riemann Sphere
. We set
to be the stereographic projection given as by:
![( 2ζ ∣ζ∣2 - 1) φ(ζ) = --2----,--2---- ∈ ℂ × ℝ = ℝ3, for ζ ∈ ℂ, and φ(∞) = (0,0,1) = N ∈ S2 ⊂ ℝ3 ∣ζ∣ + 1 ∣ζ∣ + 1](/img/revistas/ruma/v46n2/2a07265x.png)
Notice that in the class , if
, then
. If
, then
, hence
, then
.
Via a composition of two maps, we define the diffeomorphism from
onto
: for
we set,
![( - ) ( - ) Ψ([u]) = φ (L(u)) = 2a b, ∣a∣2 - ∣b∣2 = 2ab, 1 - 2∣b∣2 ∈ S2](/img/revistas/ruma/v46n2/2a07277x.png)
Considering the curve in
with
as in (3.2) above, and setting
, it can be verified that
is given by,
(3.4)
(3.5)
Notice then that parametrizes a straight line
in
. Hence the curve
![Ψ([q(t)]) = φ(L(q(t)))](/img/revistas/ruma/v46n2/2a07287x.png)
is an arc of a circle in (not necessarily a great circle) contained in the plane in
that contains both the line
, in the equatorial plane, and the North Pole
, in
. It can be verified that this plane has unit normal vectors given by:
![± (cos(β)cos(α) , cos(β) sin( α), sin(β))](/img/revistas/ruma/v46n2/2a07293x.png)
where .
3.2.2. Some observations on the curves and
in
. Let
, where
, and let
runs over a great circle in
if and only if
(equivalently
).
runs over a great circle in
(
has parameter
).
varies continuously with the parameter
.
starts at
and returns to that point exactly for
.
has constant speed
in
.
has constant speed
in
.
3.2.3. The curves in
. Lets give explicit values of the "parameters"
and
that define
and
(according to formulas (3.2) and (3.3)), so that for
, the curves
and
join the point
to
and
respectively.
Suppose that the distances from to
and
in
are
and
respectively (with
).
By means of some rotation of the sphere we may suppose that
is in the plane generated by
and
, as in figure 1 below, and we have,
.
Figure 1: We suppose that is in the plane generated by
and
For we set
so that
.
We have to choose the values and
that define
. This is equivalent to chose
via the change of variables given by the equations
![-r z- √ -2----2 cos(β) = λ , sin(β) = λ , with λ = r + z .](/img/revistas/ruma/v46n2/2a07355x.png)
The parameters and
are shown in figure 1, with the only restriction that the vector
![⃗n = (cos(β) cos(α) , cos(β) sin(α), sin(β))](/img/revistas/ruma/v46n2/2a07358x.png)
is orthogonal to a plane that contains
and
.
The parameter is determined after choosing
and
so that the short arc joining
and
, in the intersection of the plane
with the sphere
as in figure 1, has length
equal to
, from where the value of
is drawn.
[1] Durán, C. E., Mata-Lorenzo, L. E. and Recht, L., Metric geometry in homogeneous spaces of the unitary group of a C-algebra: Part I-minimal curves, Adv. Math. 184 No. 2 (2004), 342-366. [ Links ]
[2] Whittaker, E. T. "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies", Cambridge University Press, London 1988. [ Links ]
Esteban Andruchow
Instituto de Ciencias,
Universidad Nacional de Gral. Sarmiento,
J. M. Gutierrez (1613) Los Polvorines, Argentina
eandruch@ungs.edu.ar
Luis E. Mata-Lorenzo
Universidad Simón Bolívar, Apartado 89000,
Caracas 1080A, Venezuela
lmata@usb.ve
Lázaro Recht
Universidad Simón Bolívar, Apartado 89000,
Caracas 1080A, Venezuela, and
Instituto Argentino de Matemática, CONICET, Argentina
recht@usb.ve
Alberto Mendoza
Universidad Simón Bolívar, Apartado 89000,
Caracas 1080A, Venezuela
jacob@usb.ve
Alejandro Varela
Instituto de Ciencias,
Universidad Nacional de Gral. Sarmiento,
J. M. Gutierrez (1613) Los Polvorines, Argentina
avarela@ungs.edu.ar
Recibido: 23 de marzo de 2006
Aceptado: 7 de agosto de 2006