A characterization of orthogonal vector fields over ${\bf W^*}$-algebras of type ${\bf I_2}$

0.18 MB·English

by  G. D. Lugovaya

A CHARACTERIZATION OF ORTHOGONAL VECTOR
FIELDS OVER W∗-ALGEBRAS OF TYPE I2
3
1 G.D.LUGOVAYAANDA.N.SHERSTNEV*
0
2
Abstract. In the paper we give a characterization of a w∗-continuous
n orthogonal vector field F over an W∗-algebra N of type I2 in terms of
a reductions F onthecenter of N. As anapplicationitisobtained aproof
J of the assertion that an arbitrary w∗-continuous orthogonal vector field

1 overaW∗-algebraoftypeI2 isstationary.
2

]
A
Introduction
O
. In [1],[2] Masani studied the integration with respect to orthogonal vector
h
t measures defined on rings of the sets. These works obtained an extension in
a
numerouspublicationsincontextofthe noncommutativemeasuretheory. Some
m
settings of problems and approaches to their solutions may be found in [3],
[
[4,§31]. One of intrinsic problem in this subject is the problem of characteri-
1 zation of Hilbert-valued linear mappings of W∗-algebra preserving the orthog-
v onality property (so-called orthogonal vector fields). In some sense there is a
8 standard (well known) result for the commutative W∗-algebra (see for instance
6
[4, Theorem 31.19]).
7
4 In the paper we give a characterizationof a w∗-continuous orthogonalvector
1. field F overan W∗-algebraN of type I2 in terms of reductions F on the center
0 of N. As anapplication it is obtained a proofof the assertionthat anarbitrary
3 w∗-continuousorthogonalvectorfieldoveraW∗-algebraoftypeI isstationary.
2
1
: 1. Preliminaries
v
i
X Let A be a W∗-algebra, and Apr, Asa, Aun denote the sets of orthogonal
r projections, selfadjoint, unitaries elements in A, respectively. We denote by
a
rp(x)therangeprojectionofx∈A+. Itistheleastprojectionofallprojections
p∈Apr such that px=x. In the paper we shall mainly deal with W∗-algebras
of type I . It is known that any W∗-algebra of type I can be represented in
2 2
the form N = M⊗M where M is a commutative W∗-algebra and M is the
2 2
1991 Mathematics Subject Classification. Primary46L10,46L51.
Key words and phrases. W∗-algebra,orthogonal vectorfield,stationaryvector field.
1
2 G.D.LUGOVAYAANDA.N.SHERSTNEV
algebra of all 2×2 matrices over C. So the elements of N are matrices in the
x x
form of (x )= 11 12 where x ∈M. We denote by I and 1 the units
ij x x ij
21 22
(cid:18) (cid:19)
ofalgebrasN andMrespectively;{ε } denotethematrixunitofalgebra
ij i,j=1,2
N in the form
1 0 0 1 0 0 0 0
ε = , ε = , ε = , ε = .
11 0 0 12 0 0 21 1 0 22 0 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
It is known [5, Prop. 1.18.1] that a commutative W∗-algebra M may be
realized as algebra L∞(Ω,ν) of all essentially bounded locally ν-measurable
functions on a localizable measure space (Ω,ν) (i. e. Ω is direct sum of fi-
nite measure spaces, see [6]). In this case, the Banach space L1(Ω,ν) is the
predual of L∞(Ω,ν). Now we shall identify M with L∞(Ω,ν). In this case the
characteristic functions
1, if ω ∈π,
π(ω)≡χ (ω)= π ⊂Ω.
π 0, if ω 6∈π,
(cid:26)
correspondtoprojectionsπ ∈Mpr. (Weusethesameletterπtodesignatethree
objects: a projection in Mpr, a ν-measurable set in Ω and the characteristic
function of this set.)
Accordingly,the W∗-algebraN =M⊗M is realizedasthe algebraof2×2-
2
matrices (x ), x ∈L∞(Ω,ν) with the predual
ij ij
N∗ =L1(Ω,ν)⊗(M2)∗ =L1(Ω,ν,M2) (1)
where L1(Ω,ν,M ) is the Banach space of all M -valued Bochner ν-integrable
2 2
functions on Ω [5, Prop. 1.22.12]. Elements of the space M on the right-hand
2
side of (1) can be viewed as density matrices. In other words, elements ϕ of N∗
are the matrices
ϕ ϕ
ϕ= 11 12 , ϕ ∈L1(Ω,ν), (2)
ϕ ϕ ij
21 22
(cid:18) (cid:19)
and the action of ϕ on an element a = (a ), a ∈ L∞(Ω,ν), is given by the
ij ij
equality
ϕ(a)= Tr[(ϕ )(a )]dν = (ϕ a +ϕ a +ϕ a +ϕ a )dν.
ij ij 11 11 12 21 21 12 22 22
Z Z
Ω Ω
In this case
ϕ(πε )= ϕ dν, π ∈Mpr, i,j =1,2,1
ij ji
Z
π
1Weusethenotations oftypeaε12≡(cid:18) 00 a0 (cid:19), a∈M.
ORTHOGONAL VECTOR FIELDS OVER W∗-ALGEBRAS OF TYPE I2 3
i. e. ϕ = d ϕ((·)ε ) is the Radon-Nikodym derivative of the charge π →
ij dν ji
ϕ(πε ), π ∈Mpr, with respect to measure ν.
ji
Let H be a complex Hilbert space. A bounded linear map F :A→H is said
to be an orthogonal vector field (OVF) if
pq =0 (p,q ∈Apr)⇒hF(p),F(q)i=0. (3)
If a linear map F with the property (3) is continuous in w∗-topology on A and
weak topology on H, then the map F is referred to as w∗-continuous OVF. It
should be noted that a w∗-continuous OVF F is bounded, so that it is an OVF
[7, Corollary 3]. A w∗-continuous OVF F is said to be stationary [3], if there
exist two functionals ϕ,ψ ∈A+ such that
∗
hF(x),F(y)i=ϕ(y∗x)+ψ(xy∗), x,y ∈A. (4)
Given OVF F :A→H assign a positive linear functional ̺∈A∗,
̺(x)≡hF(x),F(1)i, x∈A (5)
(̺∈A+ as soon as F is w∗-continuous). The following equalities hold
∗
kF(x)k2 =̺(x2), x∈Asa, (6)
1
RehF(x),F(y)i= ̺(xy+yx), x,y ∈Asa. (7)
2
Equality (6) follows from the spectral theorem with regard to (3), and the fol-
lowing computation gives (7):
1
RehF(x),F(y)i= [hF(x+y),F(x+y)−F(x−y),F(x−y)i]
4
1 1
= [̺((x+y)2)−̺((x−y)2)]= ̺(xy+yx), x,y ∈Asa.
4 2
It is easily seen that for a stationary OVF the next equality is valid
̺=ϕ+ψ. (8)
2. Orthogonal vector fields over W∗-algebras of type I2
Let N = M⊗M be a W∗-algebra of type I , H be a Hilbert space over
2 2
C and F : N → H be a H-valued OVF. By the proof of Theorem 31.6 [4], the
mappings F :M→H given by the equalities
ij
F (a)≡F(aε ), a∈M, i,j =1,2, (9)
ij ij
are OVFs over M. Moreover,
2
F(x)= F (x ), x=(x )∈N.
ij ij ij
i,j=1
X
4 G.D.LUGOVAYAANDA.N.SHERSTNEV
SincethecenterofN isisomorphictoalgebraM,wecanconsidertheorthogonal
vector fields F as reductions F on the center of N.
ij
Proposition 1. Let F :N →H be a OVF and F are defined in (9). Then
ij
(i) hF (a),F (b)i=hF (a),F (b)i=0, a,b∈M,
11 22 12 21
(ii) kF (a)k2+kF (a)k2 =kF (a)k2+kF (a)k2, a∈M,
12 21 11 22
(iii) hF (a),F (b)i=hF (b∗a),F (1)i, a,b∈M, i,j,k,l=1,2,
ij kl ij kl
(iv) hF (π),F (1)i=hF (1),F (π)i, hF (π),F (1)i=hF (1),F (π)i,
12 11 22 21 21 11 22 12
π ∈Mpr.
Proof. Let us verify the second equality in (i). In view of the spectral theorem
it suffices to prove the assertion in case a = σ, b = τ are projections in Mpr.
Let us first assume that στ =0. We have
0 σ 1 σ ωσ 0 0 1 τ ωτ
= ω , = ω ,
0 0 4 ωσ σ τ 0 4 ωτ τ
(cid:18) (cid:19) ω=±1,±i (cid:18) (cid:19) (cid:18) (cid:19) ω (cid:18) (cid:19)
X X
(10)
where in right-hand sides of (10) there are linear combinations of projections
from N. It now follows in view of (3) that στ =0⇒hF (σ),F (τ)i=0.
12 21
Now we suppose that σ =τ. By the equalities
0 σ 1 0 σ i 0 −i σ
= + ,
0 0 2 σ 0 2 i σ 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
0 0 1 0 σ i 0 i σ
= +
σ 0 2 σ 0 2 −i σ 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
we have
0 σ 0 0
4hF (σ),F (σ)i=4 F ,F
12 21 0 0 σ 0
(cid:28) (cid:18) (cid:19) (cid:18) (cid:19)(cid:29)
2
0 σ 0 −i σ 0 i σ
= F + F ,F
σ 0 i σ 0 −i σ 0
(cid:13) (cid:18) (cid:19)(cid:13) (cid:28) (cid:18) (cid:19) (cid:18) (cid:19)(cid:29)
(cid:13)(cid:13) 0 σ (cid:13)(cid:13) 0 i σ 0 −i σ 0 σ
−i(cid:13) F (cid:13),F +i F ,F
σ 0 −i σ 0 i σ 0 σ 0
(cid:28) (cid:18) (cid:19) (cid:18) (cid:19)(cid:29) (cid:28) (cid:18) (cid:19) (cid:18) (cid:19)(cid:29)
2 2
0 σ 0 −i σ
= F − F
σ 0 i σ 0
(cid:13) (cid:18) (cid:19)(cid:13) (cid:13) (cid:18) (cid:19)(cid:13)
(cid:13)(cid:13) 0 σ(cid:13)(cid:13) (cid:13)(cid:13) 0 −i σ (cid:13)(cid:13) 0 −i σ 0 σ
+i(cid:13) F (cid:13) ,F(cid:13) (cid:13)+ F ,F .
σ 0 i σ 0 i σ 0 σ 0
(cid:26)(cid:28) (cid:18) (cid:19) (cid:18) (cid:19)(cid:29) (cid:28) (cid:18) (cid:19) (cid:18) (cid:19)(cid:29)(cid:27)
ORTHOGONAL VECTOR FIELDS OVER W∗-ALGEBRAS OF TYPE I2 5
In addition (see (6),(7)),
2 2
0 −i σ 0 −i σ
F = F ,F(I)
i σ 0 i σ 0
(cid:13) (cid:18) (cid:19)(cid:13) * (cid:18) (cid:19) ! +
(cid:13) (cid:13)
(cid:13) (cid:13) σ 0 0 σ 2
(cid:13) (cid:13) = F ,F(I) = F ,
0 σ σ 0
(cid:28) (cid:18) (cid:19) (cid:29) (cid:13) (cid:18) (cid:19)(cid:13)
(cid:13) (cid:13)
The expression in braces vanishes: (cid:13) (cid:13)
(cid:13) (cid:13)
0 σ 0 −i σ
{...}=2Re F ,F
σ 0 i σ 0
(cid:28) (cid:18) (cid:19) (cid:18) (cid:19)(cid:29)
0 σ 0 −i σ 0 −i σ 0 σ
=̺ + =0,
σ 0 i σ 0 i σ 0 σ 0
(cid:20)(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)(cid:21)
and the second equality in (i) is established. The equality hF (a),F (b)i =
11 22
0 (a,b∈M) is obvious in view of the spectral theorem.
Equality (ii) follows from computation (with regard to (i))
kF (a)k2+kF (a)k2 =kF (a)k2+kF (a∗)k2 =kF (a)+F (a∗)k2
12 21 12 21 12 21
2
0 a 0 a 0 a
= F ,F = F ,F(I)
a∗ 0 a∗ 0 a∗ 0
(cid:28) (cid:18) (cid:19) (cid:18) (cid:19)(cid:29) * (cid:18) (cid:19) ! +
=hF (a∗a),F (1)i+hF (a∗a),F (1)i=kF (a)k2+kF (a)k2.
11 11 22 22 11 22
Weusedhere(forG=F )thefollowingpropertyoftheOVFovercommutative
21
W∗-algebra M (it follows easily from the spectral theorem):
kG(a)k2 =kG(a∗)k2, a∈M.
Let us establish (iii). We first note that representations (10) give
στ =0 (σ,τ ∈Mpr)⇒hF (σ),F (τ)i=hF (σ),F (τ)i=0, i=1,2. (11)
ii 12 ii 21
InviewofthespectraltheoremandboundednessoflinearmapsF itissufficient
ij
to establish (iii) for elements a,b in the form
a= µ π , b= λ π (λ ,µ ∈C, π ∈Mpr, π π =0 (t6=s)).
t t t t t t t t s
t t
X X
(Here the sums are finite and π = 1.) Taking into account (i) and (11), we
t
t
have P
hF (a),F (b)i= µ λ hF (π ),F (π )i= µ λ hF (π ),F (1)i
ij kl t s ij t kl s t t ij t kl
t,s t
X X
=hF ( µ λ π ),F (1)i=hF (b∗a),F (1)i.
ij t t t kl ij kl
t
X
6 G.D.LUGOVAYAANDA.N.SHERSTNEV
(iv). We show first that
hF (aπ)+F (a∗π),F (π)i=hF (π),F (aπ)+F (a∗π)i, a∈M, π ∈Mpr.
12 21 11 22 12 21
(12)
0 aπ
Define f =F12(aπ)+F21(a∗π)=F a∗π 0 . By virtue of (iii) and (7)
(cid:18) (cid:19)
hf,F (π)i=hf,F(I)i−[hf,F (π)i+hF (π),fi]+hF (π),fi
11 22 22 22
0 aπ 0 0
=hf,F(I)i−2Re F a∗π 0 ,F 0 π +hF22(π),fi
(cid:28) (cid:18) (cid:19) (cid:18) (cid:19)(cid:29)
=̺(aπε +a∗πε )−̺((aπε +a∗πε )·πε +πε ·(aπε +a∗πε ))
12 21 12 21 22 22 12 21
+hF (π),fi=hF (π),fi,
22 22
and (12) is established. By setting a=1 and a=i1 in (12), we have
hF (π)+F (π),F (π)i=hF (π),F (π)+F (π)i, (13)
12 21 11 22 12 21
ihF (π)−F (π),F (π)i=−ihF (π),F (π)−F (π)i. (14)
12 21 11 22 12 21
Multiplying both sides of (14) by i and taking into account (13), we have (iv).
This proves the proposition. (cid:3)
Proposition 2. Let F (i = 1,2) be OVFs satisfying conditions (iii) and (iv)
ij
of Prop. 1. Then equality (12) holds.
Proof. Because the linear mappings F are bounded, it is sufficient to prove
ij
(12) for finite sums in the form a = λ π where λ ∈ C,π ∈ Mpr, π π =
s s s s s t
s
0 (s6=t), πs =1. In view of (iii) anPd (iv) we have
s
P
hF (aπ)+F (a∗π),F (π)i= [λ hF (ππ ),F (1)i+λ hF (ππ ),F (1)i]
12 21 11 s 12 s 11 s 21 s 11
s
X
= [λ hF (1),F (ππ )i+λ hF (1),F (ππ )i]
s 22 21 s s 22 12 s
s
X
=hF (π),F ( λ ππ )i+hF (π),F ( λ ππ )i
22 21 s s 22 12 s s
s s
X X
=hF (π),F (a∗π)+F (aπ)i.
22 21 12
The proposition follows. (cid:3)
Theorem 3. Let N = M⊗M be a W∗-algebra of type I , F : M → H be
2 2 ij
H-valued OVFs (w∗-continuous OVFs) with the properties (i)−(iv) of Prop. 1.
Then the linear mapping F :N →H defined by the equalities
F(aε )≡F (a), a∈M, i,j =1,2,
ij ij
is the OVF (respectively, w∗-continuous OVF).
ORTHOGONAL VECTOR FIELDS OVER W∗-ALGEBRAS OF TYPE I2 7
Proof. Since F is w∗–continuous if and only if F are w∗-continuous, it is suf-
ij
ficient to consider the case when F be OVFs. We have only to verify the
ij
equality
pq =0 ⇒ hF(p),F(q)i=0, p,q∈Npr.
We use the canonical representation projections in N. Specifically, every pro-
jection r ∈Npr can be expressed in the form [8, Lemma 1]
r =π ⊕π +p(a,v,π), (15)
1 2
where
π 0
π ⊕π ≡ 1 , π ∈Mpr,
1 2 0 π i
2
(cid:18) (cid:19)
aπ vπ(a(1−a))1/2
p(a,v,π)≡ , π ∈Mpr,v ∈Mun.
v∗π(a(1−a))1/2 (1−a)π
(cid:18) (cid:19)
In addition, π ≤1−π (i=1,2), a∈M, 0≤a≤1,rp a(1−a))=1. Now we
i
present p,q ∈Npr in the form of (15):
p=τ ⊕τ +p(a,v,τ ), q =σ ⊕σ +p(b,w,σ ),
1 2 3 1 2 3
where τ ,σ ∈ Mpr, 0 ≤ a,b ≤ 1, v,w ∈ Mun. From pq = 0 it follows [8,
i i
Lemma 2] that
τ σ =τ τ =σ σ =τ σ =τ σ =0, wπ =−vπ, bπ =(1−a)π, (16)
i i i 3 i 3 3 i i 3
where i=1,2, π ≡σ τ . Denoting for brevity κ(x)≡(x(1−x))1/2 we have
3 3
F(p)=F (τ +aτ )+F (vκ(a)τ )+F (v∗κ(a)τ )+F (τ +(1−a)τ ),
11 1 3 12 3 21 3 22 2 3
F(q)=F (σ +bσ )+F (wκ(b)σ )+F (w∗κ(b)σ )+F (σ +(1−b)σ ).
11 1 3 12 3 21 3 22 2 3
Takingthe scalarproductof vectorsF(p) and F(q) with regardto (i), (iii), (iv)
in Prop. 1 and equalities (16) we have
hF(p),F(q)i=hF (aπ),F ((1−a)π)i−hF (aπ),F (vκ(a)π)i
11 11 11 12
−hF (aπ),F (v∗κ(a)π)i+hF (vκ(a)π),F ((1−a)π)i
11 21 12 11
−kF (vκ(a)π)k2+hF (vκ(a)π),F (aπ)i
12 12 22
+hF (v∗κ(a)π),F ((1−a)π)i−kF (v∗κ(a)π)k2
21 11 21
+hF (v∗κ(a)π),F (aπ),i−hF ((1−a)π),F (vκ(a)π)i
21 22 22 12
−hF ((1−a)π),F (v∗κ(a)π)i+hF ((1−a)π),F (aπ)i.
22 21 22 22
ByProp. 1(ii),thefirstandthelastsummandsontherighthand-sideofobtained
equality are mutually annihilated with the fifth and the eights ones. Grouping
8 G.D.LUGOVAYAANDA.N.SHERSTNEV
the remained summands and setting c =vaκ(a), d=v(1−a)κ(a) we have by
(12)
hF(p),F(q)i=[hF (vκ(a)π)+F (v∗κ(a)π),F (aπ)i
12 21 22
−hF (aπ),F (vκ(a)π)+F (v∗κ(a)π)i]
11 12 21
+[hF (vκ(a)π)+F (v∗κ(a)π),F ((1−a)π)i
12 21 11
−hF ((1−a)π),F (vκ(a)π)+F (v∗κ(a)π)i]
22 12 21
=[hF (cπ)+F (c∗π),F (π)i−hF (π),F (cπ)+F (c∗π)i]
12 21 22 11 12 21
+[hF (dπ)+F (d∗π),F (π)i−hF (π),F (dπ)+F (d∗π)i]=0.
12 21 11 22 12 21
The proof is complete. (cid:3)
3. The stationarity of OVFs over W∗-algebras of type I2
As an application of Theorem 3 we will show that every w∗-continuous OVF
overanW∗-algebraoftype I is stationary. Let N =M⊗M be a W∗-algebra
2 2
oftypeI ,F :N →H beaw∗-continuousOVFandthefunctional̺begivenby
2
(5). Letusobservesomepropertiesofreductions(10). Withtheabovenotations
of Section 1 we have
̺(πε )=hF(πε ),F(I)i=hF(πε ),F(ε )i=hF (π),F (1)i, (17)
ii ii ii ii ii ii
where i=1,2, π ∈Mpr. Therefore,
d
ρ = hF (·),F (1)i, i=1,2. (18)
ii dν ii ii
Similarly,
d
ρ = hF (·),F (1)+F (1)i, i,j =1,2 (i6=j). (19)
ij dν ji 11 22
In addition there are defined functions r ∈L1(Ω,ν)+ such that
ij
d
hF (π),F (1)i= r dν, r = hF (·),F (1)i, i,j =1,2. (20)
ij ij ij ij dν ij ij
Z
π
In this connection
r =̺ (i=1,2), r +r =̺ +̺ a. e. (21)
ii ii 12 21 11 22
(The second equality in (21) follows from Prop. 1(ii).)
If F is stationary there exist (see (4)) ϕ,ψ ∈N+ such that
∗
hF(x),F(y)i=ϕ(y∗x)+ψ(xy∗), x,y ∈N. (22)
ORTHOGONAL VECTOR FIELDS OVER W∗-ALGEBRAS OF TYPE I2 9
In this case (8) is satisfied. Let (ϕ ) be the matrix of ϕ (see (2)). We have
ij
ϕ dν =ϕ(πε )=ϕ(ε (πε )ε )=hF((πε )ε ),F(ε )i
21 12 11 12 22 12 22 11
Z
π
=hF(πε ),F(ε )i=hF (π),F (1)i, π ∈Mpr.
12 11 12 11
Thus,
d d
ϕ = hF (·),F (1)i, ϕ =ϕ = hF (·),F (1)i.
21 12 11 12 21 21 22
dν dν
(The equality ϕ =ϕ follows from the positivity of ϕ.) Further,
12 21
(ϕ −ϕ )dν =ϕ(πε −πε )=ϕ(πε ε∗ )−ϕ(πε∗ ε )
11 22 11 22 12 12 12 12
Z
π
=ϕ(πε ε∗ )+ψ(πε∗ ε )−̺(πε∗ ε )
12 12 12 12 12 12
=hF(πε∗ ),F(πε∗ )i−̺(πε )
12 12 22
=kF(πε )k2−̺(πε )=hF (π),F (1)i− ̺ dν
21 22 21 21 22
Z
π
= (r −̺ )dν.
21 22
Z
π
(The function r is defined by (20).) Putting φ≡ϕ we have
21 11
φ ϕ ̺ −φ ̺ −ϕ
ϕ= 12 , ψ = 11 12 12 . (23)
ϕ φ+̺ −r ̺ −ϕ r −φ
12 22 21 12 12 21
(cid:18) (cid:19) (cid:18) (cid:19)
It should be noted that all elements of matrices ϕ and ψ, with the exception of
φ, are defined as the Radon-Nikodym derivatives associated with the fields F .
ij
The function φ∈L1(Ω,ν)+ necessarily satisfies inequalities
max{0,r (ω)−̺ (ω)}≤φ(ω)≤min{̺ (ω),r (ω)} a. e., (24)
21 22 11 21
|ϕ (ω)|≤[φ(ω)(φ(ω)+̺ (ω)−r (ω))]1/2 a. e., (25)
12 22 21
|̺ (ω)−ϕ (ω)|≤[(ρ (ω)−φ(ω))(r (ω)−φ(ω))]1/2 a. e. (26)
12 12 11 21
Theorem 4. Let N = M⊗M be a W∗-algebra of type I . Then every w∗-
2 2
continuous OVF F :N →H is stationary.
The following lemma gives conditionally ahead of time proof of the theorem.
Lemma 5. If the function φ ∈ L1(Ω,ν) satisfies inequalities (24) – (26), then
the equality (22) holds when ϕ and ψ are defined by matrices (23).
10 G.D.LUGOVAYAANDA.N.SHERSTNEV
Proof. As F is linear and w∗-continuous (and by Prop. 3(iii)) we must verify
only that the following equalities hold
hF(πε ),F(ε )i=ϕ(ε (πε ))+ψ((πε )ε ), π ∈Mpr, i,j,k,l∈{1,2}.
ij kl lk ij ij lk
They are easily verified directly. For example, in view of (17),(8),(9) and by
Prop. 1(iv) we have
hF(πε ),F(ε )i= ̺ dν = (ϕ +ψ )dν =ϕ(πε )+ψ(πε )
ii ii ii ii ii ii ii
Z Z
π π
=ϕ(ε (πε ))+ψ((πε )ε ), i=1,2,
ii ii ii ii
hF(πε ),F(ε )i=hF (π),F (1)i=hF (1),F (π)i
11 12 11 12 11 12
=hF (π),F (1)i= ϕ dν =ϕ(πε )
21 22 12 21
Z
π
=ϕ((πε )ε )+ψ(ε (πε )).
21 11 11 21
Similarly we can verify remaining equalities, and the lemma follows. (cid:3)
Lemma 6. Any OVF over the factor of type I is stationary.
2
Proof. Let F : M →H be a OVF. Without loss of generality, we may assume
2
that kF(I)k=1. Let us denote for brevity F =F(ε ). By virtue of standard
ij ij
argumentsconnectedwithunitaryinvariance,itissufficienttoconsiderthecase
when the matrix ̺=(̺ ) (here it is scalar) is diagonal:
ij
̺ 0
̺= 11 , ̺ +̺ =1, ̺ ≥0.
0 ̺ 11 22 ii
22
(cid:18) (cid:19)
Also note that
r =kF k2, r +r =1.
ij ij 12 21
Case 1: the rank of matrix ̺ equals 1. Let̺ =1,̺ =0(the samearguments
11 22
can be used if ̺ =0,̺ =1). By the equality ̺ =0 it follows that {F } is
11 22 12 ij
theorthogonalsystemofvectorsinH. Nowweseethatequality(22)issatisfied
with
r 0 r 0
ϕ= 21 , ψ = 12
0 0 0 0
(cid:18) (cid:19) (cid:18) (cid:19)