vol7_pp21-29. 191KB Jun 04 2011 12:06:15 AM

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The Electronic Journal of Linear Algebra.

A publication of the International Linear Algebra Society. Volume 7, pp. 21-29, February 2000.

ISSN 1081-3810. http://math.technion.ac.il/iic/ela

ELA

POLAR DECOMPOSITIONUNDER PERTURBATIONS

OF THESCALAR PRODUCT

GUSTAVO CORACHy, ALEJANDRA MAESTRIPIERIz,

AND DEMETRIO STOJANOFF x

Dedicated to our friend and teacher Angel Rafael Larotonda on his sixtieth birthday

Abstract. Let A be a unital C

-algebra with involution

represented in a Hilbert space H, Gthe group of invertible elements of A, U the unitary group of A,G

s the set of invertible selfadjoint elements of A,Q = f" 2 G :"

2 = 1

g the space of reections and P = Q\U. For any positivea2Gconsider thea-unitary groupUa=fg 2G:a

,1 g

a=g

,1

g, i.e., the elements which are unitary with respect to the scalar producth;ia =ha;ifor ; 2 H. If denotes the map that assigns to each invertible element its unitary part in the polar decomposition, it is shown that the restriction j

U a :

U a

! U is a dieomorphism, that(U a

\Q) = P, and that (U

a \G

s) = U

a \G

s=

fu2G:u=u =

u ,1 and

au=uag:

Keywords.Polar decomposition, C

-algebras, positive operators, projections.

AMSsubject classications.Primary 47A30, 47B15

1. Introduction. If A is the algebra of bounded linear operators in a Hilbert

spaceH, denote byGthe group of invertible elements ofA. EveryT 2Gadmits two

polar decompositions

T =U 1

P 1=

P 2

U 2

whereU 1

;U

2are unitary operators (i.e. U

i =

U ,1 i ) and

P 1

;P

2are positive operators

(i.e hP i

;i 0 for every 2 H). It turns out that U 1 =

U 2,

P 1 = (

T

T) 1=2 and P

2= ( TT

)1=2. We shall call U =U

1= U

2the unitary part of

T. Consider the map

:G!U (T) =U

where U is the unitary group of A. If G

+ denotes the set of all positive invertible

elements of A, then every A 2 G

+ denes an inner product h ; i

A on

H which is

equivalent to the originalh; i; namely

h;i A=

hA;i (;2H):

Received by the editors on 28 September 1999. Accepted for publication on 16 January 2000. Handling editor: Chandler Davis.

yInstituto Argentino de Matematica, Saavedra 15, Piso 3, 1083 Buenos Aires, Argentina ([email protected]). Partially supported by Fundacion Antorchas, CONICET (PIP 4463/96), Universidad de Buenos Aires (UBACYT TX92 and TW49) and ANPCYT (PICT97-2259)

zInstituto de Ciencias, UNGS, Roca 851, San Miguel, Buenos Aires, Argentina ([email protected])

xDepartamento de Matematica, UNLP, calles 1 y 50, 1900 La Plata, Argentina ([email protected]). Partially supported by CONICET (PIP 4463/96), Universidad de Buenos Aires (UBACYT TW49) and ANPCYT (PICT97-2259)

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ELA

22 Polar decomposition under perturbations

Every X 2 A admits anA-adjoint operator X

A, which is the unique

Y 2 A such

that

hX;i A=

h;Yi

A (

;2H):

It is easy to see thatX

A =

A ,1

X

A. Together with the denition of

A one gets

the sets ofA-Hermitian operators

A h A=

fX2A:X

A=

Xg=fX2A:AX=X

Ag; A-unitary operators

U A=

fX 2G:X

A= X

,1

g=fX2A:AX ,1=

X

Ag

andA-positive operators

G + A=

fX2A h A

\G:hX;i A

0 82H g

As in the \classical" case, i.e. A=I, we get two polar decompositions for eachT 2G

T =V 1

R 1=

R 2

V 2

withV i

2U A,

R i

2G + A,

i= 1;2 and as beforeV 1=

V 2=

V. Thus, we get a map

A: G!U

A

;

A(

T) =V ; T 2G:

This paper is devoted to a simultaneous study of the maps A (

A 2 G

+), the way

that

U A

; G

h A

; G

+ A

intersect

U B

; G

h B

; G

+ B

for dierentA;B2G

+ and the intersections of these sets with

Q=fS2A:S 2=

Ig and P

A=

fS2Q:S

A= Sg

(reections and A-Hermitian reections of A). The main result is the fact that, for

everyA;B 2G +,

A

j UB :

U B

!U A

is a bijection. The proof of this theorem is based on the form of the positive solutions of the operator equationXAX=BforA;B2G

+. This identity was rst studied by

G. K. Pedersen and M. Takesaki [10] in their study of the Radon-Nykodym theorems in von Neumann algebras. As a corollary we get a short proof of the equality

A(

Q\U B) =

P A

for everyA;B 2G

+, which was proven in [1] as a C

-algebraic version of results of

Pasternak-Winiarski [8] on the analyticity of the mapA7!P M A , where

P M A is the

A

-orthogonal projection on the closed subspaceMofH. We include a parametrization

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G.Corach,A.Maestripieri,andD.Stojano 23

2. Preliminaries.

LetAbe a unital C

-algebra,

G=G(A) the group of

invert-invertible elements ofA and G s=

G s(

A) the set of positive selfadjoint elements of A. LetQ=Q(A) =f"2G:"

2= 1

gbe the space of reections and

P =P(A) =Q\G

the space of orthogonal reections, also called the Grassmann manifold ofA.

Eachg2Gadmits two polar decompositions

g=u=u

exercise of functional calculus shows thatu=u

0. We shall say that

uistheunitary partofg. Observe that in the decompositiong=u(resp. g=u

0) the components ; u(resp. u,

0) are uniquely determined, for instance, if

u=

,1 is a unitary element with positive spectrum: (

is a bration with very rich geometric properties (see [11], [2] and the references therein). We are interested in the way that the 0

,1( u) = G

+

u =uG

+ intersect

the base space of a similar bration induced by a dierent involution. More precisely, eacha2G

+ induces a C

involution on

A, namely

+ induces the inner product h;i

-algebra with

this involution and with the normkk

a associated to h;i

+, consider the unitary group U

a corresponding to the involution

#a:

We shall study the restriction

Ua

and the way that dierent U a

; U

b are set in G.

Moreover, we shall also consider thea-hermitian part ofG,

G

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ELA

and the intersections of these sets whenavaries inG

+. The reader is referred to [7]

and [5] for a discussion of operators which are hermitian for some inner product. Observe that each a2 G

+ induces a bration

a :

G! U

a with bers

homeo-morphic toG +

a. This paper can be seen in some sense as a simultaneous study of the

brations a,

a2G +.

Let us mention that, from an intrinsic viewpoint, U

a can be identied with U.

Indeed, consider the map' a:

A!Agiven by '

a( b) =a

,1=2 ba

1=2 ( b2A):

(2) Then'

a( U) =U

a, '

a( G

s) = G

s a and

' a(

G +) =

G + a, since

' a: (

A;)!(A;# a) is an

isomorphism of C-algebras. We are concerned with the way in which the base space

and bers of dierent brations behave with respect to each other.

3. The polar decomposition.

In [10], Pedersen and Takesaki proved a

tech-nical result which was relevant for their generalization of the Sakai's Radon-Nikodym theorem for von Neumann algebras [11]. More precisely, they determined the unique-ness and existence of positive solutions of the equation THT =K forH;K positive

bounded operators in a Hilbert space. We need a weak version of their result, namely when H;K are positive invertible operators. In this case it is possible to give an

explicit solution.

Lemma 3.1 ([10]). IfH;K are positive invertible bounded operators in a Hilbert

space, the equation

THT=K

(3)

has a unique solution, namely T=H ,1=2(

H 1=2

KH 1=2)1=2

H ,1=2

:

Proof. Multiply (3) at left and right byH

1=2 and factorize

H 1=2

THTH 1=2= (

H 1=2

TH 1=2)2

:

Then we get the equation (H 1=2

TH

1=2)2 = H

1=2 KH

1=2

: Taking (positive) square

roots and using the invertibility ofH

1=2we get the result. 2

Returning to the map:G!U, consider the ber ,1(

u) =fu:2G +

g. In

order to compare the bration with

a, the following is the key result Theorem 3.2. Let a2G

+. Then, for every

u2U the ber ,1(

u) intersects U

a at a single point, namely a

,1=2( a

1=2 uau

,1 a

1=2)1=2 a

,1=2

u: In other words, the

restrictionj U

a: U

a

!U is a homeomorphism.

Proof. Ifg =u2U a then

a ,1

g

a=g

,1 is equivalent to a

,1 u

,1 a=u

,1

,1 ;

so, after a few manipulations,

a=uau

,1 :

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By Pedersen and Takesaki's result, there is a unique2G

+ which satises equation

(4) for xed a 2 G +,

u 2 U, namely = a ,1=2(

a 1=2

uau ,1

a 1=2)1=2

a ,1=2

: Thus,

(j Ua)

,1: U !U

a is given by

(j U

a) ,1(

u) =a ,1=2(

a 1=2

uau ,1

a 1=2)1=2

a ,1=2

u

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ELA

which obviously is a continuous map. 2

Leta2G

+ and consider the involution #

a dened in equation (1). It is natural

to look at those reections " 2 Q which are #

a-orthogonal, i.e the so called #a

-Grassmann manifold ofA. Let us denote this space by

P

In [8], Pasternak-Winiarski studied the behavior of the orthogonal projection onto a closed subspace of a Hilbert space when the inner product varies continuously. Note that we can identify naturally the space of idempotents q with the reections

of Q via the ane map q 7! " = 2q,1, which also maps the space of orthogonal

projections ontoP. Based on [8], a geometric study of the space Q is made in [1],

where the characterization(P a) =

Pis given (proposition 5.1 of [1]). In the following

proposition we shall give a new proof of this fact by showing that the homeomorphism

j

P U. Therefore the formula given in equation (5)

for the inverse ofj U

a extends the formula given in proposition 5.1 of [1] for ( j

P a)

,1,

since they must coincide onP. Proposition 3.3. Let a 2 G

+. Then (P

a) =

(Q\U a) =

P: Therefore j

P a:

P a

!P is a homeomorphism.

Proof. By the previous remarks, we just need to show that (P a) =

since the unitary part of"corresponding to both right and left polar decompositions

coincide, we get ,1=

Thus, applying the continuous functional calculus (see e.g. [9]) to the selfadjoint elementa

1=2

3.1. Positive parts. In order to complete the results on the relationship

be-tween polar decomposition and inner products, consider the complementary map of the decompositiong=u, namely

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Of course, there is another \complementary map", namelyg7!(g

g)1=2

correspond-ing to the decompositiong=u0. We shall see that for everya 2G

+, the restriction

+ j

G + a :G

+ a

!G

+

is a homeomorphism. Indeed, given2G

+, consider the polar decomposition a=

u, with 2G

+ and u

2U. Then a

,1=u, so+(a,1) = anda,1 2G

+ a,

sincea,1(a,1)a=a,1and the spectrum(a,1)

R

+. Note that

(+ j

G + a)

,1() =a,1+(a);

which is clearly a continuous map. An interesting rewriting of the above statement is the following:

Proposition3.4. If Aisaunital C

-algebraanda;2G +

,thenthere existsa uniqueu2U suchthat au2G

+: Proof.Indeed, ifg= (

+ j

G + a)

,1() 2G

+

a andu=(g), thenu=g 2G

+ a means

exactly thatau2G +.

2

It is worth mentioning that x 2 G is the unique positive solution of

Pedersen-Takesaki equationxax =b if and only if a1=2xb,1=2

2U. Changing a;bby a 2;b,2

respectively, we can write Lemma 3.1 as follows:

Proposition3.5. If Aisaunital C

-algebraanda;b2G +

,thenthereexistsa uniquex2G

+

suchthataxb2U:

3.2. Products of positive operators.

The map :G+

G

+

!U given by

(a;b) =axb=a(a,1(ab2a)1=2a,1)b= (ab2a)1=2a,1b; a;b 2G

+;

is not surjective: in fact, the image of consists of those unitary elements which can be factorized as a product of three positive elements. On one side (a;b) =axb2U

is the product of three elements of G+. On the other side, if axb

2 U then by

Pedersen-Takesaki's resultx is the unique positive solution ofxa2x=b,2.

It is easy to show that,12Ucan not be decomposed as a product of four positive

elements. See [12] and [13] for a complete bibliography on these factorization prob-lems. See [3] for more results on factorization of elements ofGand characterizations ofPn=

fa

1:::an:ai 2G

+

g, at least in the nite dimensional case.

3.3. Parametrization of the solutions of Pedersen-Takesaki equations.

Given a;b 2 G

+, denote by m = jb

1=2a1=2 j = (a

1=2ba1=2)1=2. Then the set of all

solutions of the equationxax=bis

fa

,1=2 m " a,1=2 : "

2Q and "m=m"g:

In fact,xax=bif and only if (a1=2xa1=2)2=m2and the set of all solutions ofx2=c2

forc2G + is

fc" : "

2= 1 and "c=c"

g:The singular case, which is much more

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4. Intersections and unions. For any selfadjointc2Awe shall consider the

relative commutant subC-algebra

A

c and, analogously G

The spaceG

shas a deep relationship with

Q(in [4] there is a partial description

of it). Here we only need to notice that the unitary part of any c2G

s also belongs

to P. Indeed, if is the polar decomposition of c, then = c = c

=

. By

the uniqueness of the unitary part, = = Theorem 4.1. LetAaunital C

Proof.By the previous remarks, ifb2G s

\U a and

b=is its polar decomposition,

then2P and

By the uniqueness of the positive solution,=

,1 and, since 2 G

+, this means

that = 1. Thus a = a and then 2 P(A

a). Conversely, if

2 P(A

+. Then easy computations show that

1. U

We shall give two proofs of item 4: First proof: (G

+, then its spectrum

(x)S 1

\R +=

f1g; on the other

side, x is normal with respect to the involutions # a, so

x is a normal element such

that(x) =f1gand it must bex= 1.

Letb2G

+. Recall that the map '

b dened in (2) changes the usual involution

by #b and also all the corresponding spaces (e.g ' Proof.Notice thatU

d =

( and the same happens for G s and

c and the other two

identities. 2

Then we can generalize the results above for any paira;b2G

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1. U

Proof.Use Proposition 4.1, Remark 4.2 and Lemma 4.3. 2

In the following proposition we describe the set of elements ofAwhich are unitary

(resp. positive, Hermitian) for some involution #a ( a 2 G

+). We state the result

without proof.

Proposition4.5. IfA isaunitalC

-algebra,the following identitieshold: [

whereG +

The following example shows that there is no obvious spectral characterization of these subsets ofG: if xis nilpotent, then 1 +x does not belong to any of them but (1 +x) =f1gR;R

+ and the circle S

1.

4.1. Final geometric remarks.

All subsets ofAstudied in this paper have a

rich structure as dierential manifolds. The reader is referred to [6] and [2] for the case ofU and to [4] (and the references therein) forQ;P;G

sand G

+. The map '

adened

in equation (2) is clearly a dieomorphism which allows to get all the information on

U

a from that available on U;G

s ;G

+, respectively. The main results of the

paper say that the map is a dieomorphism betweenU a and

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