Klyachko’s Theorem in Semi-finite von Neumann Algebras

Klyachko’s Theorem in Semi-finite von Neumann Algebras

By

TetsuoHarada^∗

Abstract

LetAbe a von Neumann algebra andAbe the set of allτ-measurable operators.

For positive elementsAandBinAwe prove that Z

Kµs(A+B)ds≤ Z

Iµs(A)ds+ Z

Jµs(B)ds,

where µ(·) denotes the generalized s-number andI,J, andKare on an analogue of the Klyachko list.

§1. Introduction

Recently, Klyachko [11] has shown that the possible eigenvaluesα,β,γof Hermitiann×nmatricesA,B, andC=A+B are characterized by a certain list of inequalities togather with the trace equality

γ_i =

α_i+ β_i. If we set α= (α₁, . . . , α_n),β = (β₁, . . . , β_n), and γ = (γ₁, . . . , γ_n) in decreasing order, then this list of inequalities is of the form

(1)

k∈K

γ_k ≤

i∈I

α_i+

j∈J

β_j

for certain subsetsI,J,Kof the same cardinalityr, which are on the Klyachko list, withr < n(i.e. This list includes 3n−3 inequalities: α₁≥α₂≥ · · · ≥α_n, β₁≥β₂≥ · · · ≥β_n,γ₁≥γ₂≥ · · · ≥γ_n.). We setλ(I) = (i_r−r, . . . , i₂−2, i₁−1)

Communicated by H. Okamoto. Received June 1, 2006.

2000 Mathematics Subject Classification(s): Primary 47B15; Secondary 46L10, 47A05.

Key words: von Neumann algebra, singular number, spectral scale.

∗9-16-201 Hakozaki 1-chome, Higashi-ku, Fukuoka 812-0053, Japan.

e-mail: [email protected]

for I ={i₁, . . . , i_r}. Then (I, J, K) is on the Klyachko list exactly when the Littlewood-Richardson coefficient C_λ(I)^λ(K)_λ(J) is positive (cf. [6], [7], [8]). From the solution of the saturation conjecture by Knutson-Tao [12], one can show that this list is exactly what was conjectured by Horn [10].

There is a long history to find necessary conditions. Arrange the eigenval- ues of a Hermitian matrixX in decreasing order:λ(X) = (λ₁(X), . . . , λ_n(X)).

For example, in 1912 H. Weyl found

λ_i+j−1(A+B)≤λ_i(A) +λ_j(B) wheneveri+j−1≤n.

Some other inequalities were found in 1949 by Ky Fan:

r k=1

λ_k(A+B)≤^r

k=1

λ_k(A) + r k=1

λ_k(B) for anyr < n.

It is easy to check ({i},{j},{i+j−1}) and ({1, . . . , r},{1, . . . , r},{1, . . . , r}) are on the Klyachko list.

There are some generalizations for the above inequality (1). Friedland [4] proved the inequality (1) for positive compact operators. Bercovici-Li [2]

proved the inequality (1) for self-adjoint operators inII₁ factors. Their meth- ods are simple and useful, but we think that the same method cannot apply for the continuous infinite case, since they used discrete and finite properties respectively. In this paper, we will prove the inequality (1) by another method for any positive τ-measurable operators. Of course, this result includes the infinite case (in particular, the continuous infinite case). And also, when A is a finite von Neumann algebra, we can prove the inequality (1) not only for positive operators but also for bounded self-adjoint operators. More precisely, Theorem 3.1 shows

J0

µ_s(A+B)ds≤

J1

µ_s(A)ds+

J2

µ_s(B)ds

for positive operatorsA,B. LetC,D be bounded self-adjoint operators in A, there exists a positive numberxsuch thatC+xI≥0,D+xI ≥0. Therefore, we get

J0

µ_s(C+D+ 2xI)ds≤

J1

µ_s(C+xI)ds+

J2

µ_s(D+xI)ds.

Sinceµ_s(C+xI) =λ_s(C)+xandm(J₀) =m(J₁) =m(J₂)<∞(see Definition 2.4), the above inequality implies

J0

λ_s(C+D)ds≤

J1

λ_s(C)ds+

J2

λ_s(D)ds.

§2. Notation

In this section, we will collect some definitions and basic facts. Throughout the paper, letAbe a von Neumann algebra on a Hilbert spaceHwith a faithful normal semi-finite traceτ. LetAdenote the set of allτ-measurable operators.

Definition 2.1 (τ-measurable operators). A densely-defined closed op- erator A affiliated with A is said to be τ-measurable if for each ε > 0 there exists a projection E in A such thatE(H)⊂ D(A) and τ(1−E)≤ε. (Here, D(A) denotes the domain ofA.)

Let A be a closed densely-defined operator affiliated with A. Let |A| = _∞

0 λ de_λ(|A|) be the spectral decomposition. Then it is easy to check thatA is τ-measurable if and only if τ(1−e_λ(|A|))<∞ forλ large enough (cf. [3], [14]).

Definition 2.2 (The generalizeds-number, The spectral scale). For a self-adjoint element A∈A, the distribution functiond_s(A) is defined by

d_s(A) =τ(e_(s,∞)(A)),

where e_(s,∞)(A) is the spectral projection of A corresponding to the interval (s,∞).

The generalized s-number is defined by

µ_t(A) = inf{s:d_s(|A|)≤t} (0< t <∞) while the spectral scale is defined by

λ_t(A) = inf{s:d_s(A)≤t} (0< t <∞).

The above definitions correspond to the decreasing rearrangement of the eigenvalues of |A|, andA respectively. (Whene_[0,∞)(A) =∞, λ_t(A) does not necessarily correspond to the decreasing rearrangement of the eigenvalues ofA.

Therefore, we use the spectral scale only for elements in a finite von-Neumann algebra.) Of course, for positive operators, both definitions are the same.

If A is a τ-measurable operator, then we have d_s(|A|) < ∞ for s large enough and lim_s→∞d_s(|A|) = 0 as noted before. Moreover, µ_t(A), λ_t(A) are non-increasing and right continuous on (0,∞). See [3], [15] for detailed prop- erties of the above functions.

Definition 2.3 (Continuous flag).

Let{e_s}_s∈[0,∞)be a net of projections inAsatisfying the following condi- tions.

(i) τ(e_s)≤s

(ii) Ifs≤t, then, e_s ≤e_t

Then, we call{e_s}_s∈[0,∞)a continuous flag onA. Definition 2.4 (Admissibility).

Let J be a measurable set in [0, τ(1)). (Here, m denotes Lebesgue mea- sure.) For a continuous flag {e_s}, Ω_J({e_s}, τ) is the set of projections in A defined by

Ω_J({e_s}, τ) ={p∈Aproj.;τ(p) =m(J), τ(1−p) =m(J^c), τ(p∧e_s)≥ m{J ∩[0, s]}(for all s∈[0, τ(1)) )}. Let J₀, J₁, J₂ be Lebesgue measurable sets in [0, τ(1)) such that m(J₀) = m(J₁) =m(J₂) andm(J₀^c) =m(J₁^c) =m(J₂^c). A triple (J₀, J₁, J₂) is called ad- missible for continuous flags ({e_s},{f_s},{g_s}), if the following condition holds:

Ω_J0({e_s}, τ)∩(Ω_Jc

1({f_s}, τ))^⊥∩(Ω_Jc

2({g_s}, τ))^⊥=∅,

where (Ω_J1^c({f_s}, τ))^⊥={1−p;p∈Ω_J1^c({f_s}, τ)}. Of course, whenτ(1)<∞, the above requirements for the complement of sets are automatically met.

Definition 2.5. ForA∈Awe set SG(A) = inf

ξ∈H\{0}

||Aξ||

||ξ|| . Proposition 2.1. Let A,B be operators in A. 1. ||Aξ|| ≥ SG(A)· ||ξ|| forξ∈ H.

2. SG(AB)≥ SG(A)· SG(B).

3. SG(A) =||A⁻¹||⁻¹, ifA is invertible.

Proof. 1. follows immediately from the definition ofSG(·).

2., 3. From the definition, we have SG(AB) = inf

ξ∈H\{0}

||ABξ||

||ξ||

≥ inf

ξ∈H\{0}

SG(A)· ||Bξ||

||ξ||

=SG(A)· SG(B),

SG(A) = inf

ξ∈H\{0}

||Aξ||

||A⁻¹(Aξ)||

= inf

ξ∈H\{0}

||A⁻¹(Aξ)||

||Aξ||

−1

=

sup

ξ∈H\{0}

||A⁻¹ξ||

||ξ||

₋₁

(byA(H) =H).

Definition 2.6. LetAbe an operator inAandE be a projection inA. Then the restriction of SG(A) for a projectionE (denote itSG(A)_E) is

SG(A)_E= inf

ξ∈E(H)\{0}

||Aξ||

||ξ|| .

For this restrictionSG(·)_E, an anologue of Proposition 2.1 is valid.

Proposition 2.2. Let A,B be operators in AandE,F be projections in A.

1. ||Aξ|| ≥ SG(A)_E· ||ξ|| forξ ∈E(H).

2. SG(AB)_E≥ SG(A)_E· SG(B)_E.

3. IfE(H)is an invariant subspace ofAandAEis invertible as an element in B(E(H)). thenSG(A)_E =||(EAE)⁻¹||⁻¹_E .Here,||·||E is the usual operator norm in B(E(H)).

4. SG(E)_E= 1, if E =\ 0.

5. SG(A)_E≥ SG(A)_F, ifE ≤F.

Proof. 1., 2. are trivial (as in the proof of Proposition 2.1).

3. We compute

SG(A)_E= inf

ξ∈E(H)\{0}

||Aξ||

||ξ||

= inf

ξ∈E(H)\{0}

||EAEξ||

||ξ||

= inf

ξ∈E(H)\{0}

||EAEξ||

||(EAE)⁻¹EAEξ||

=

sup

ξ∈E(H)\{0}

||(EAE)⁻¹EAEξ||

||EAEξ||

₋₁

=

sup

ξ∈E(H)\{0}

||(EAE)⁻¹ξ||

||ξ||

₋₁

=||(EAE)⁻¹||⁻¹_E . 4. SG(E)_E= inf_{ξ∈E(H)\{0}}^||Eξ||_||ξ|| = 1.

5.

SG(A)_E= inf

ξ∈E(H)\{0}

||Aξ||

||ξ||

≥ inf

ξ∈F(H)\{0}

||Aξ||

||ξ||

=SG(A)_F.

§3. Main Theorem Proposition 3.1. ForA∈A we have

µ_t(A) = inf{||AE||;E is a projection in Awith τ(1−E)≤t}

= sup{SG(A)_E;E is a projection inA withτ(E)≥t}.

Proof. The first equality is shown in [3]. We show the second equality.

Let us denote the supremum in this proposition by α. Ifα= 0, then we have µ_t(A)≥α. So we can assumeα >0. Then, for a sufficiently smallε >0, there exists a projectionE withτ(E)≥tsuch that

0< α−ε <SG(A)_E.

Ifξ ∈E(H)∩e_[0,α−ε](|A|)(H),||ξ||= 1, then we have (A^∗Aξ|ξ) =||Aξ||²≥ SG(A)²_E>(α−ε)², (A^∗Aξ|ξ) =|| |A|e_[0,α−ε](|A|)ξ||²≤(α−ε)². ThereforeE∧e_[0,α−ε](|A|) = 0, and we compute

E=E−E∧e_[0,α−ε](|A|)

∼E∨e_[0,α−ε](|A|)−e_[0,α−ε](|A|)

≤1−e_[0,α−ε](|A|)

=e_{(α−ε,∞)}(|A|).

We thus get

τ(e_{(α−ε,∞)}(|A|))≥τ(E)≥t, α−ε≤µ_t(A).

On the other hand, set E =e_[µt(A),∞)(|A|). If µ_t(A) = 0, then we have µ_t(A)≤α. So we can assumeµ_t(A)>0. Asτ(E)≥tandSG(A)_E ≥ SG(|A|)_E (Let A = u|A| be the polar decomposition. Since E(H) ⊂ |A|(H), we have SG(u)_E= 1.), we get

α≥ SG(|A|)_E

=||(E|A|E)⁻¹||⁻¹_E

=

[µt(A),∞)

λ⁻¹de_λ(|A|) ⁻¹

E

≥µ_t(A).

Proposition 3.2. Let A be a positive operator in A. Let J be a mea- surable set in[0, τ(1)), and pbe a projection inA.

1. If τ(p∧e_(µs(A),∞)(A))≥m{J∩[0, s]}, then µ_m{J∩[0,s]}(pAp)≥µ_s(A).

2. If τ(R(pe_(µs(A),∞)(A))) ≤m{J ∩[0, s]}, then µ_m{J∩[0,s]}(pAp)≤ µ_s(A).

Here, R(pe_(µs(A),∞)(A))denotes the range projection of pe_(µs(A),∞)(A).

Proof. 1. Ifµ_s(A) = 0, then it is trivial. So we can assume µ_s(A)>0.

From Proposition 2.2, we compute

µ_m{J∩[0,s]}(pAp)≥ SG(pAp)_{p∧e(µs(A),∞)(A)}

≥ SG(p)_{p∧e(µs(A),∞)(A)}· SG(A)_{p∧e(µs(A),∞)(A)}

≥ SG(A)_{e(µs(A),∞)(A)}

=

(µs(A),∞)

λ⁻¹de_λ(A) ⁻¹

e_(µs(A),∞)(A)

≥µ_s(A).

2. Consider the set ofτ-measurable operators pAp. SincepApis in A,

µ_t(pAp) = inf{||pApE||;Eis a projection inpApwithτ(p−E)≤t}. Therefore,

µ_m{J∩[0,s]}(pAp)≤ ||pAp(p− R(pe_(µs(A),∞)(A)))||

=||pAp(p∧e_[0,µs(A)](A))||

=||pA(p∧e_[0,µs(A)](A))||

≤ ||A(p∧e_[0,µs(A)](A))||

=||Ae_[0,µs(A)](A)(p∧e_[0,µs(A)](A))||

≤ ||Ae_[0,µs(A)](A)||

≤µ_s(A).

Proposition 3.3. LetAbe a positive operator inAandJ be a Lebesgue measurable set in[0, τ(1)). Letpbe a projection inAwithm(J) =τ(p). Then,

J

µ_m{J∩[0,s]}(pAp)ds= _τ(p)

0

µ_s(pAp)ds=τ(pAp).

Proof. It suffices to prove that µ_m{J∩[0,s]}(pAp) and µ_s(pAp) have the same distribution function. At first we claim that

m{s∈J;µ_m{J∩[0,s]}(pAp)> t}=m{J∩[0, α]}(t >0) withα= sup{s∈J;µ_m{J∩[0,s]}(pAp)> t}.

Letube an element inJ such thatµ_m{J∩[0,u]}(pAp)> t, then m{s∈J;µ_m{J∩[0,s]}(pAp)> t} ≥m{J∩[0, u]}. (since µ_tis decreasing fors≤u,µ_m{J∩[0,s]}(pAp)> t)

From the definition ofαthere exists{α_j}in{s∈J;µ_m{J∩[0,s]}(pAp)> t} such that α_j α.

Therefore, we have m{s∈J;µ_m{J∩[0,s]}(pAp)> t} ≥ m{J ∩[0, α_j]}(for allj). Taking the lim_j→∞of the both sides, we get

m{s∈J;µ_m{J∩[0,s]}(pAp)> t} ≥m{J∩[0, α]}. Conversely, from the definition ofα, we obviously have

m{s∈J;µ_m{J∩[0,s]}(pAp)> t} ≤m{J∩[0, α]}.

Finally, we will prove thatm{J∩[0, α]}=m{s;µ_s(pAp)> t}. Clearly, m{J∩[0, α_j]} ≤m{s;µ_s(pAp)> t}.

(since µ_{m{J∩[0,αj]}}(pAp)> t) So we get

m{J∩[0, α]} ≤m{s;µ_s(pAp)> t}. On the other hand, we set

β = sup{s;µ_s(pAp)> t}. Sinceµ_s is decreasing,m{s;µ_s(pAp)> t}=β

If β =∞, thenα=τ(p) =m(J) =∞. Therefore, we get the conclusion. So we can assumeβ <∞. When we set

f(x) =m{J∩[0, x]} (x∈[0, τ(1)) ),

f is a measurable function on [0, τ(1)). Futhermore,f is a continuous finction on [0, τ(1)).

Indeed, if x_j →x, then

j→∞lim m{J∩[0, x_j]}= lim

j→∞

_τ(1)

0

χ_J∩[0,xj] dm

= _τ(1)

0

j→∞lim χ_J∩[0,xj] dm

(by the dominated convergence theorem)

=m{J∩[0, x]}.

Sincef(0) = 0,f(τ(1)) =m(J) =τ(p),f attains any value in [0, τ(p)].

Therefore, for allε >0, there existsxsuch that m{J∩[0, x]}=β−ε.

When we sety= sup{J∩[0, x]},

{J∩[0, x]}={J∩[0, y]}.

From the definition ofy, there exists{y_j}in{J∩[0, x]}such thaty_jy. We get

j→∞lim m{J∩[0, y_j]}=m{J∩[0, x]}=β−ε.

Asm{J∩[0, y_j]} ≤β−ε, we haveµ_{m{J∩[0,yj]}}(pAp)> t. Therefore we get m{J∩[0, α]} ≥ lim

j→∞m{J∩[0, y_j]}=β−ε (from the definition ofα, y_j≤α) The proof is complete since εis arbitrary.

Proposition 3.4. Let A be a positive operator in A. Let J be a mea- surable set, and pbe a projection inA.

1. If p∈Ω_J({e_(µs(A),∞)(A)}, τ), then τ(pAp)≥

Jµ_s(A)ds.

2. If p∈(Ω_Jc({e_(µs(A),∞)(A)}, τ))^⊥, then τ(pAp)≤

Jµ_s(A)ds.

Proof. 1. Proposotions 3.2 and 3.3 show the result.

2. From the Definition of (Ω_Jc({e_(µs(A),∞)(A)}, τ))^⊥,

τ((1−p)∧e_(µs(A),∞)(A))≥m{J^c∩[0, s]} (for alls∈[0, τ(1)) ).

By substracting the both sides froms, we have

s−τ((1−p)∧e_(µs(A),∞)(A))≤s−m{J^c∩[0, s]} (for all s∈[0, τ(1)) ).

We note

s−τ((1−p)∧e_(µs(A),∞)(A))

≥τ(e_(µs(A),∞)(A))−τ((1−p)∧e_(µs(A),∞)(A))

=τ(e_(µs(A),∞)(A)−(1−p)∧e_(µs(A),∞)(A))

=τ((1−p)∨e_(µs(A),∞)(A)−(1−p))

=τ(R(pe_(µs(A),∞)(A))),

and

s−m{J^c∩[0, s]}

=m{[0, s]} −m{J^c∩[0, s]}

=m{J∩[0, s]}. Using Propositions 3.2 and 3.3, we get

τ(pAp)≤

J

µ_s(A)ds.

Theorem 3.1. Let A and B be positive operators in A. Let J₀, J₁ andJ₂ be measurable sets withm(J₀) =m(J₁) =m(J₂)andm(J₀^c) =m(J₁^c) = m(J₂^c). If a triple (J₀, J₁, J₂) is admissible for ({e_{(µs(A+B),∞)}(A + B)}, {e_(µs(A),∞)(A)},{e_(µs(B),∞)(B)}), then

J0

µ_s(A+B) ds≤

J1

µ_s(A)ds+

J2

µ_s(B)ds.

Proof. Ifp∈Ω_J0({e_{(µs(A+B),∞)}(A+B)}, τ)∩(Ω_Jc

1({e_(µs(A),∞)(A)}, τ))^⊥

∩(Ω_Jc

2({e_(µs(B),∞)(B)}, τ))^⊥, then τ(p(A+B)p)≥

J0

µ_s(A+B)ds(by Proposition 3.4 ).

On the other hand, sincep∈(Ω_J1^c({e_(µs(A),∞)(A)}, τ))^⊥, Proposition 3.4 shows τ(pAp)≤

J1

µ_s(A)ds.

Similarly, for an operatorB we get τ(pAp)≤

J2

µ_s(B)ds.

Therefore, we obtain

J0

µ_s(A+B)ds≤τ(p(A+B))

=τ(pA) +τ(pB)

≤

J1

µ_s(A)ds+

J2

µ_s(B)ds.

Lemma 3.1. ForJ= [0, r]we have

Ω_Jc({e_(µs(A),∞)(A)}, τ)^⊥={p∈A_proj.;τ(p) =r}.

Proof. Letpbe a projection withτ(p) =r. It is enough to prove that τ((1−p)∧e_(µs(A),∞)(A))≥m{[0, s]∩[0, r]^c}.

Ifs≤r, it is trivial. Whenr < s <∞, we get

τ(R(e_(µs(A),∞)(A)p)) =τ(R(pe_(µs(A),∞)(A)))

≤τ(p)

=r

=m{[0, s]∩[0, r]}. By substracting the both sides froms, we get the conclusion.

From Theorem 3.4 and Lemma 3.1 we obtain

Ω_[0,r]({e_{(µs(A+B),∞)}(A+B)}, τ)∩(Ω_[0,r]c({e_(µs(A),∞)(A)}, τ))^⊥

∩(Ω_[0,r]c({e_(µs(B),∞)(B)}, τ))^⊥= Ω_[0,r]({e_{(µs(A+B),∞)}(A+B)}, τ).

WhenAhas no minimal projection, Ω_[0,r]({e_{(µs(A+B),∞)}(A+B)}, τ) is not an empty set, because e_{(µr(A+B),∞)}(A+B) is inclueded. Since we can always embed Ainto A⊗L^∞_([0,1];dt), we get the inequality

_r

0

µ_s(A+B)ds≤ _r

0

µ_s(A)ds+ _r

0

µ_s(B)ds (for all r∈[0, τ(1)) ).

This inequalty was found by Ky Fan when A = M_n(C). For a general von Neumann algebra with a faithful semifinite normal trace several proofs are known (cf. [3], [9]).

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