A Schur-Horn type theorem for II$_1$ factors

0.16 MB·

by  M. Argerami

A SCHUR-HORN THEOREM IN II FACTORS
1
7
0
M.ARGERAMIANDP.MASSEY
0
2
n Abstract. Given a II1 factor M and a diffuse abelian von Neumann subal-
gebraA⊂M,weproveaversionoftheSchur-Horntheorem,namely
a
J EA(UM(b))σ-sot={a∈Asa: a≺b}, b∈Msa,

2 where≺denotesspectralmajorization,EAtheuniquetrace-preservingcondi-
1 tionalexpectationontoA,andUM(b)theunitaryorbitofbinM. Thisresult

isinspiredbyarecentproblemposedbyArvesonandKadison.
]
A
O
.
h 1. Introduction
t
a In 1923, I. Schur [18] proved that if A∈M (C)sa (i.e., A is selfadjoint) then
m n
k k
[
α↓ ≤ β↓, k =1,...,n,
j j
4 j=1 j=1
X X
v
0 with equality when k = n (denoted α ≺ β), where α= diag(A) ∈Rn, β = λ(A) ∈
2 Rn the spectrum (counting multiplicity) of A, and α↓,β↓ ∈ Rn are obtained from
1 α,β by reordering their entries in decreasing order.
4 In 1954, A. Horn [12] proved the converse: given α,β ∈ Rn with α ≺ β, there
0
exists a selfadjoint matrix A ∈ M (C) such that diag(A) = α, λ(A) = β. Since
6 n
0 every selfadjoint matrix is diagonalizable, the results of Schur and Horn can be
/ combinedin the following assertion: if D denotes the diagonalmasa in Mn(C) and
h
E is the compression onto D , then
t D
a
m (1) ED({UMβU∗ :U ∈Mn(C) unitary})={Mα ∈D: α≺β},
: where M is the diagonal matrix with the entries of α in the main diagonal.
v α
This combination of the two results, commonly known as Schur-Horn theorem,
i
X
has played a significant role in many contexts of matrix analysis: although simple,
r vector majorization expresses a natural and deep relation among vectors, and as
a
such it has been a useful tool both in pure and applied mathematics. We refer to
the books [5, 14] and the introductions of [6, 16] for more on this.
During the last 25 years,severalextensions of majorizationhave been proposed
by, among others, Ando [1] (to selfadjoint matrices), Kamei [13] (to selfadjoint
operators in a II factor), Hiai [10, 11] (to normal operators in a von Neumann
1
algebra), and Neumann [16] (to vectors in ℓ∞(N)). With these generalizations at
hand, it is natural to ask about extensions of the Schur-Horn theorem.
2000 Mathematics Subject Classification. Primary46L99,Secondary 46L55.
Key words and phrases. Majorization,diagonalsofoperators,Schur-Horntheorem.
M.ArgeramisupportedinpartbytheNaturalSciences andEngineeringResearchCouncil of
Canada.
P.MasseysupportedinpartbyCONICETofArgentinaandaPIMSPostdoctoralFellowship.
1
2 M.ARGERAMIANDP.MASSEY
In [16], Neumann developedhis extensionof majorizationwith the goalof using
it to prove a Schur-Horn type theorem in B(H) in the vein of previous works in
convexity(seetheintroductionin[16]fordetailsandbibliography). Otherversions
of the Schur-Horn theorem have been considered in [3] and [6]. It is interesting to
note that the motivation in [16] comes from geometry, in [3] comes from the study
of frames on Hilbert spaces, while in [6] it is of an operator theoretic nature.
In[6]ArvesonandKadisonproposedthe studyofaSchur-Horntype theoremin
thecontextofII factors,whichareforsuchpurposethemostnaturalgeneralization
1
of full matrix algebras. They proveda Schur type theorem for II factors and they
1
posed as a problem a converse of this result, i.e. a Horn type theorem. In this
note we prove a Schur-Horn type theorem that is inspired by Arveson-Kadison’s
conjecture (Theorem 3.4).
2. Preliminaries
Throughout the paper M denotes a II factor with normalized faithful normal
1
traceτ. We denoteby Msa, M+, U , the setsofselfadjoint,positive,andunitary
M
elements of M, and by Z(M) the center of M. Given a ∈ Msa we denote its
spectral measure by pa. The characteristic function of the set ∆ is denoted by
1 . For n ∈ N, the algebra of n×n matrices over C is denoted by M (C), and
∆ n
its unitary group by U . By dt we denote integration with respect to Lebesgue
n
measure. To simplify terminology, we will refer to non-decreasing functions simply
as “increasing”;similarly, “decreasing” will be used instead of “non-increasing”.
Besides the usual operator norm in M, we consider the 1-norm induced by the
trace, kxk = τ(|x|). As we will be always dealing with bounded sets in a II
1 1
factor, we can profit from the fact that the topology induced by k·k agrees with
1
theσ-strongoperatortopology. Becauseofthiswewillexpressourresultsinterms
of σ-strong closures although our computations are based on estimates for k·k .
1
σ-sot
For X ⊂ M, we shall denote by X and X the respective closures in the norm
and in the σ-strong operator topology.
2.1. Spectral scale and spectral preorders. Thespectral scale [17]ofa∈Msa
is defined as
λ (t)=min{s∈R: τ(pa(s,∞))≤t}, t∈[0,1).
a
The function λ : [0,1) → [0,kak] is decreasing and right-continuous. The map
a
a7→λ is continuous with respect to both k·k and k·k , since [17]
a 1
(2) kλ −λ k ≤ka−bk, kλ −λ k ≤ka−bk a, b∈Msa,
a b ∞ a b 1 1
wherethenormsontheleftarethoseofL∞([0,1],dt)andL1([0,1],dt)respectively.
We say that a is submajorized by b, written a≺ b, if
w
s s
λ (t) dt≤ λ (t) dt, for every s∈[0,1).
a b
Z0 Z0
If in addition τ(a)=τ(b) then we say that a is majorized by b, written a≺b.
Thesepreordersplayanimportantroleinmanypapers(amongthemwemention
[8, 10, 11, 13]), and they arise naturally in severalcontexts in operator theory and
operator algebras: some recent examples closely related to our work are the study
of Young’s type [9] and Jensen’s type inequalities [2, 6, 7, 10].
A SCHUR-HORN THEOREM IN II1 FACTORS 3
Theorem 2.1 ([10]). Let a, b ∈ Msa. Then a ≺ b (resp. a ≺ b) if and only if
w
τ(f(a)) ≤ τ(f(b)) for every convex (resp. increasing convex) function f : J → R,
where J is an open interval such that σ(a), σ(b)⊆J.
If N ⊂M is a von Neumann subalgebra and b∈Msa, we denote by Ω (b) the
N
set of elements in Nsa that are majorized by b, i.e.
Ω (b)={a∈Nsa : a≺b}.
N
The unitary orbit of a∈Msa is the set U (a)={u∗au:u∈U }.
M M
Proposition 2.2. Let N ⊂ M be a von Neumann subalgebra and let E be the
N
trace preserving conditional expectation onto N. Then, for any b∈Msa,
(i) E (b)≺b.
N
(ii) kE (b)k ≤kbk .
N 1 1
σ-sot
(iii) E (U (b)) ⊂Ω (b).
N M N
Proof. (i) The map E is doubly stochastic (i.e. trace preserving, unital, and
N
positive), so this follows from [10, Theorem 4.7] (see also [6, Theorem 7.2]).
(ii) Consider the convex function f(x) = |x|. Since E (b) ≺ b, using Theorem
N
2.1 we get
kE (b)k =τ(f(E (b)))≤τ(f(b))=kbk .
N 1 N 1
(iii) By (i) and the fact that ubu∗ ≺ b for every u ∈ U , we just have to
M
prove that the set ΩN(b) is k·k1-closed. So, let (an)n∈N ⊂ ΩN(b) be such that
lim ka −ak =0 for some a∈N. Then, necessarily, a∈Nsa. By (2),
n→∞ n 1
s s s
λ (t) dt= lim λ (t) dt≤ λ (t) dt.
a n→∞ an b
Z0 Z0 Z0
Also, τ(a)=lim τ(a )=τ(b), so a≺b. (cid:3)
n n
The following result seems to be well-known, but we have not been able to find
a reference. Thus we give a sketch of a proof.
Proposition2.3. Leta∈Asa,whereA⊂MisadiffusevonNeumannsubalgebra.
Then there exists a spectral resolution {e(t)} ⊂ A with τ(e(t)) = t for every
t∈[0,1]
t∈[0,1], and such that
1
a= λ (t)de(t).
a
Z0
Proof. Since τ(1)<∞, it is enough to show the result for a≥0. By [15, Theorem
3.2] there exists a′ ∈ A with ga′(s) = τ(pa′(−∞,s]) continuous in R, and an in-
creasingleft-continuousfunctionf suchthatpa(−∞,s]=pa′(−∞,f(s)]. Although
the original statement in [15] involves a masa, only the fact that the algebra is
diffuse is needed for its proof.
Let ga†′(t)=min{s: ga′(s)≥t}, and let q(t)=pa′(−∞,ga†′(t)]. Since τ(q(t)) =
ga′(ga†′(t))=tforeveryt∈[0,1],itfollowsthat{q(t)}t∈[0,1] isacontinuousspectral
resolution. Moreover, q(ga′(t)) = pa′(−∞,t], so pa(−∞,t] = q(ga′(f(t))). As
ga′ ◦ f is increasing and right-continuous, by [15, Theorem 4.4] there exists an
increasing and left-continuous function h such that a = h (t)dq(t). Define
a a
e(t)= 1−q(1−t), and h(t) =h (1−t). Then τ(e(t)) =t, a = h(t)de(t). As h
a
is decreasing and right-continuous, it can be seen that h=λR. (cid:3)
a
R
4 M.ARGERAMIANDP.MASSEY
A spectral resolution {e(t)} as in Proposition 2.3 is called a complete flag
t∈[0,1]
for a.
3. A Schur-Horn Theorem for II factors
1
For each n ∈ N, k ∈ 1,...,2n, let {I(n)}2n be the partition of [0,1] associated
k k=1
to the points {h2−n: h=0,...,2n}.
Definition 3.1. For each n∈N and every f ∈L1([0,1]), let
2n
En(f)= i=1 2nZIi(n)f ! 1Ii(n).
X
Itis clearthateachoperatorE isalinearcontractionforbothk·k andk·k .
n 1 ∞
Given the flag {e(t)} , we write e([t ,t ]) for e(t )−e(t ). Note that, since we
t 0 1 1 0
consider e(t) diffuse, e([t ,t ])=e((t ,t ))=e([t ,t ))=e((t ,t ]).
0 1 0 1 0 1 0 1
Lemma 3.2. Let {e(t)}t∈[0,1], {Ii(n)}2i=n1, n∈N and {En}n∈N as above. Then, for
each a∈Msa,
1
(3) lim ka− E (λ )(t)de(t)k =0.
n a 1
n→∞
Z0
Proof. By continuity of the trace, we only need to check that
(4) kλ −E (λ )k −→0
a n a 1
n
in L1([0,1]). Consider first a continuous function g. By uniform continuity, kg−
E (g)k → 0. Since continuous functions are dense in L1([0,1]) and because the
n 1
operatorsE arek·k -contractiveforeveryn∈N,astandardε/3argumentproves
n 1
(4) for any integrable function. (cid:3)
Recall that D denotes the diagonal masa in M (C), and that for α ∈ Rn we
n
denotebyM thematrixwiththeentriesofαinthediagonalandzerooff-diagonal.
α
The projection E of M (C) onto D is then given by E (A) = M , where
D n D diag(A)
diag(A) ∈ Rn is the main diagonal of A. We use {e } to denote the canonical
ij
system of matrix units in M (C).
n
Lemma 3.3. Let N ⊂M be a von Neumann subalgebra, and assume E denotes
N
the unique trace preserving conditional expectation onto N. Let {p }n ⊂ Z(N)
i i=1
n
be a set of mutually orthogonal equivalent projections such that p =I. Then
i=1 i
there exists a unital *-monomorphism π :M (C)→M satisfying
n
P
(5) π(e )=p , 1≤i≤n,
ii i
(6) E (π(A))=π(E (A)), A∈M (C).
N D n
Proof. Sincetheprojectionsp areequivalentinM,foreachithereexistsapartial
i
isometry v such that v v∗ = p and v∗v = p . Let v = p , v = v∗
i1 i1 i1 i i1 i1 1 11 1 1i i1
for 2 ≤ i ≤ n and v = v v for 1 ≤ i,j ≤ n. In this way we get the standard
ij i1 1j
associatedsystemofmatrixunits{v , 1≤i, j ≤n}inM. Defineπ :M (C)→M
ij n
n
by π(A)= a v .
i,j=1 ij ij
The matrixunitrelationsimply thatπ is a*-monomorphismanditis clearthat
P
(5) is also satisfied. Moreover,
E (v )=E (p v p )=p E (v )p =δ p ,
N ij N i ij j i N ij j ij i
A SCHUR-HORN THEOREM IN II1 FACTORS 5
since p p =δ p , E (v )=E (p )=p , and p ∈Z(N). Finally, we check (6):
i j ij i N ii N i i i
E (π(A))= a E (v )= a p =π(E (A)). (cid:3)
N ij N ij ii i D
i,j i
X X
Next we state and prove our version of the Schur-Horn theorem for II factors.
1
Note the formal analogy with (1).
Theorem 3.4. Let A ⊂ M be a diffuse abelian von Neumann subalgebra and let
b∈Msa. Then
σ-sot
(7) E (U (b)) =Ω (b).
A M A
σ-sot
Proof. By Proposition 2.2, we only need to prove E (U (b)) ⊃Ω (b).
A M A
So let a∈Asa with a≺b. By Proposition 2.3,
1 1
a= λ (t)de(t), b= λ (t)df(t)
a b
Z0 Z0
where {e(t)} and {f(t)} are complete flags with τ(e(t)) = τ(f(t)) = t, e(t) ∈ A,
t∈[0,1].
Let{I(n)}2n ,n∈N,be the family ofpartitionsconsideredbeforeandletǫ>0.
i i=1
By Lemma 3.2 there exists n∈N such that
2n 2n
(8) a− α p <ǫ, b− β q <ǫ,
i i i i
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) Xi=1 (cid:13)1 (cid:13) Xi=1 (cid:13)1
(cid:13) (cid:13) (cid:13) (cid:13)
where αi = 2n Ii(n)(cid:13)(cid:13)λa(t)dt, βi =(cid:13)(cid:13)2n Ii(n) (cid:13)(cid:13)λb(t)dt, pi =(cid:13)(cid:13)e(Ii(n)), qi = f(Ii(n)), 1 ≤
i≤2n. Note thRat τ(pi)=τ(qi)=2−nR. Let α=(α1,...,α2n), β =(β1,...,β2n)∈
R2n. Fromthefactthatλ andλ aredecreasing,theentriesofαandβ arealready
a b
in decreasing order. Using that a≺b IN M, we conclude that α≺β in Rn.
By the classical Schur-Horn theorem (1), there exists U ∈U (C) such that
n
(9) E (UM U∗)=M
D β α
Consider the *-monomorphism π of Lemma 3.3 associated with the orthogonal
familyofprojections{p }2n ⊂A. Letw∈U suchthatwq w∗ =p ,i=1,...,2n,
i i=1 M i i
and put u:=π(U)w ∈U . By (6) and (9),
M
2n
E u β q u∗ = E (π(UM U∗))
A i i A β
! !
i=1
X
= π(E (UM U∗))=π(M )
D β α
2n
= α p
i i
i=1
X
Using (8), we conclude that
2n 2n
kE (ubu∗)−ak ≤ E (u(b− β q )u∗) + a− α p <2ǫ.
A 1 A i i i i
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) Xi=1 (cid:13)1 (cid:13) Xi=1 (cid:13)1
(cid:13) σ-sot(cid:13) (cid:13) (cid:13)
As ǫ was arbitrary, we obt(cid:13)ain a∈E (U (b)) (cid:13). (cid:13) (cid:13) (cid:3)
(cid:13) A M (cid:13) (cid:13) (cid:13)
σ-sot
Corollary 3.5. For each b∈M+, the set E (U (b)) is convex and σ-weakly
A M
compact.
6 M.ARGERAMIANDP.MASSEY
In [6], Arveson and Kadison posed the problem whether for b ∈ Msa, with the
notations of Theorem 3.4,
E U (b) =Ω (b).
A M A
(cid:16) (cid:17)
Since ([13])
σ-sot
U (b)=U (b) ={a∈Msa : λ =λ },
M M a b
an affirmative answer to the Arveson-Kadisonproblem is equivalent to
σ-sot
(10) E U (b) =Ω (b).
A M A
(cid:16) (cid:17)
As a description of the set Ω (b), (7) is weaker than (10), since in general
A
σ-sot σ-sot
(11) E (U (b) )⊂E (U (b)) .
A M A M
An affirmative answer to the Arveson-Kadison problem would imply equality in
(11). We think it is indeed the case,although a proofof this does not emerge from
our present methods.
Acknowledgements. WewouldliketothankProfessorsD.Stojanoff,D.Farenick,
and D. Sherman for fruitful discussions regarding the material contained in this
note. We would also like to thank the referee for severalsuggestionsthat consider-
ably simplified the exposition.
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Departmentof Mathematics,University ofRegina,ReginaSK, Canada
E-mail address: [email protected]
Departamentode Matema´tica,Universidad Nacionalde La Plata andInstituto Ar-
gentinode Matema´tica-conicet,Argentina
E-mail address: [email protected]