[PDF] - A Schur-Horn theorem in II 1 factors Mart n Argerami and Pedro PDF Document

SLIDE 1

A Schur-Horn theorem in II1 factors

Mart´ ın Argerami and Pedro Massey Florianopolis, July 2006

SLIDE 2

Majorization: x, y ∈ Rn, y ≺ x ⇐ ⇒

k

j=1

y↓

j ≤ k

j=1

x↓

j

k = 1, . . . , n − 1 and

n

j=1

yj =

n

j=1

xj. Example: if x1, . . . , xn ∈ R+, xi = 1, then (1 n, 1 n, . . . , 1 n) ≺ (x1, . . . , xn) ≺ (1, 0, . . . , 0) Characterizations: y ≺ x ⇐ ⇒ y ∈ co Sn(x). ⇐ ⇒ y = Ax, A doubly stochastic. ⇐ ⇒ tr(f(y)) ≤ tr(f(x)), for any convex function f. Schur (1923): If A ∈ Mn(C)sa, then diag(A) ≺ λ(A).

2

SLIDE 3

A. Horn (1954): If x, y ∈ Rn, and y ≺ x, then ∃ A ∈ Mn(C)sa

such that diag(A) = y, λ(A) = x. Schur-Horn (SH): Let x ∈ Rn. Then ED({UMxU ∗ : U ∈ Un}) = {My : y ≺ x}. RHS: Convex! Extensions of Majorization: Ando (1982, to s.a matrices), Kamei (1983, to s.a. operators in a finite factor), Hiai (1987-1989, to s.a. and normal operators in a von Neumann algebra), Neumann (1998, vectors in ℓ∞(N)). How about Schur-Horn in these extended settings?

3

SLIDE 4

Neumann’s answer: in ℓ∞(N), equality holds if you take closure

n both sides:

ED({UMxU ∗ : U ∈ Un})

∞ = {My : y ≺ x} ∞.

SH: a characterization, given a selfadjoint matrix A, of the set of all diagonals of selfadjoint matrices having the same eigenvalues as A. Arveson and Kadison (2005): try to extend SH with this point of view. SH in L1(H) (Arv-Kad 2005): let A ⊂ B(H) a discrete masa (i.e. o.n.b.). Let A ∈ L1(H). Then E

{UAU∗ : U ∈ U(H)}

1

= {B ∈ A ∩ L1(H) : λ(B) ≺ λ(A)}. Related: Kadison’s Carpenter Theorem (what are the diagonals

f projections?)

4

SLIDE 5

How does SH fit in II1 factors? (M, τ). Instead of λ(x) ∈ ℓ∞(N), we have λx : [0, 1) → R, λx(t) = min{s ∈ R : τ(px(s, ∞)) ≤ t}, t ∈ [0, 1). (spectral scale/singular numbers: T. Fack, D. Petz, F. Hiai). Properties of λx: decreasing, right-continuous; λx − λy∞ ≤ x − y, λx − λy1 ≤ x − y1 if x ∈ M+, λx(t) = inf{xe : e ∈ P(M), τ(1 − e) ≤ t}. τ(a) = 1 λa(t) dt. Consequence: if b ∈ UM(a)

sot, then λb = λa

5

SLIDE 6

Converse: Kamei (1983). So UM(a) = UM(a)

sot = {b ∈ Msa : λa = λb}.

Three preorders induced by the Spectral Scale Spectral domination: a b if any of the following holds: (i) λa(t) ≤ λb(t), for all t ∈ [0, 1). (ii) τ(pa(t, ∞)) ≤ τ(pb(t, ∞)), for all t. Submajorization: a ≺w b if s λa(t) dt ≤ s λb(t) dt, for every s ∈ [0, 1). Majorization: a ≺ b if a ≺w b and τ(a) = τ(b). We have a ≤ b ⇒ a b ⇒ a ≺w b ⇒ τ(a) ≤ τ(b)

6

SLIDE 7

If N ⊂ M, b ∈ Msa, let ΩN(b) = {a ∈ N sa : a ≺ b}, ΘN(b) = {a ∈ N sa : a ≺w b}. Proposition 1. Let N ⊂ M be a von Neumann subalgebra and let EN be the trace preserving conditional expectation onto N. Then, for any b ∈ Msa, (i) EN(b) ≺ b. (ii) EN(b)1 ≤ b1. (iii) EN(UM(b))

sot ⊂ ΩN(b).

(iv) If in addition b ∈ M+ then EN(CM(b))

sot ⊂ Θ+

N(b). 7

SLIDE 8

Refinements: We say that a ∈ M+ has diffuse distribution if s → τ(pa(s, ∞)) is continuous, and that a refines b if there exists an increasing right-continuous function f : [0, b] → [0, a] such that f(b) = a and pb(λ, ∞) = pa(f(λ), ∞) for every λ ∈ [0, b]. Theorem 2 (Modelling of operators). Let A ⊂ M be a masa and let a ∈ A+. Then there exists a′ ∈ A+ with diffuse distribution such that (i) a′ refines a. (ii) For each b ∈ M+ there exists an increasing left continuous function hb : [0, a′] → R+

0 such that λb = λhb(a′). Moreover,

a′ refines b if and only if hb(a′) = b. (iii) For each c ∈ M+, (a) c b if and only if hc ≤ hb; (b) c ≺w b if and only if hc(a′) ≺w hb(a′); (c) c ≺ b if and only if hc(a′) ≺ hb(a′). We say that hb(a′) is a model of b with respect to a′. Remark: EA(UM(b))

sot = EA(UM(hb(a′))) sot.

8

SLIDE 9

Approximations in · 1 W∗(a): the von Neumann subalgebra of M generated by a. Then W ∗(a)

Ψa

≃ L∞(J, ν), where ν(∆) = τ(pa(∆)), σ(a) ⊂ J, ϕν(g) =

g dν, Ψa : g(a) → g.

Also, λΨa(g) = λg. So, if g, h ∈ L∞(J, ν)+, h(a) g(a) ⇐ ⇒ h g. h(a) ≺w g(a) ⇐ ⇒ h ≺w g. h(a) ≺ g(a) ⇐ ⇒ h ≺ g. With ν diffuse, we can find partitions of J of equal measure. Proposition 3. Let g, h : [α, β] → R≥0 be increasing left-continuous functions such that g ≺ h in (L∞([α, β], ν), ϕν), and let {Ii} be an equal-measure partition of [α, β]. Let g, h ∈ R2n be given by gi = 2n

Ii

g dν, hi = 2n

Ii

h dν, 1 ≤ i ≤ 2n. Then g = g↑, h = h↑, and g ≺ h.

9

SLIDE 10

Proposition 4. Let g, h : J → R+

0 be increasing left continuous

functions. Then g ≺w h in (L∞(J, ν), ϕν) if and only if there exists an increasing left continuous function f : J → R+

0 such that

g ≺ f ≤ h. Using Proposition 4 and Theorem 2, Proposition 5. Let a, b ∈ M+. Then (i) a ≺w b if and only if there exists c ∈ M+ such that a ≺ c ≤ b. Moreover, if B ⊂ M is a masa and b ∈ B+, we can choose c ∈ B+. (ii) a b if and only if a ∈ CM(b) = {vbv∗ : v ∈ M, v ≤ 1}. Definition 6. Let {I(n)

i

}2n

i=1, n ∈ N, be a ν-dyadic partition of

[α, β]. For each n ∈ N and every f ∈ L1(ν), x ∈ [α, β], let En(f)(x) =

2n

i=1
2n
I(n)

i

f dν

1I(n)

i (x).

En is a linear contraction for both · 1 and · ∞.

10

SLIDE 11

Proposition 7. Let {I(n)

i

}2n

i=1, n ∈ N, be a ν-dyadic partition of

[α, β] and let {En}n∈N be the associated family of discrete approx-

imations. Then, for every g ∈ L1(ν),

lim

n→∞ g − En(g)1 = 0.

(1) Important idea: diameters in the partition also go to zero, up to measure zero. This makes it work for continuous functions.

11

SLIDE 12

A Topological Schur-Horn Theorem for II1 factors Recall: classical SH ED(Un(Mh)) = {Mg ∈ D : g ≺ h}, h ∈ Rn. If N ⊂ M, b ∈ Msa, let ΩN(b) = {a ∈ N sa : a ≺ b}, ΘN(b) = {a ∈ N sa : a ≺w b}. Theorem 8. Let A ⊂ M be a masa and let b ∈ M+. Then EA(UM(b))

sot = ΩA(b).

Proof. By Proposition 1, we only need to prove EA(UM(b))

sot ⊃ ΩA(b).

Let a ∈ A+ with a ≺ b. ∃a′ ∈ A+ with diffuse distribution such that it refines a, and EA(UM(b))

sot = EA(UM(hb(a′))) sot.

So we can assume that b = hb(a′) ∈ A+. Then ha(a′) = a and ha ≺ hb in (L∞(ν), ϕν). By Proposition 7, ∃ discrete approximants

a −

2n

i=1

hi pi

1

< ǫ,

b −

2n

i=1

gi pi

1

< ǫ where gi = 2n

Ii

ha dν, hi = 2n

Ii

hb dν, 1 ≤ i ≤ 2n.

12

SLIDE 13

Then, by in Proposition 3, g = g↑, h = h↑, and g ≺ h. By the classical Schur-Horn theorem there exists U ∈ Un(C) such that ED(UMhU ∗) = Mg Use U to obtain u, with EA

u

2n

i=1

hi pi

u∗
=

2n

i=1

gi pi Since u(b − 2n

i=1 gi pi)u∗1 < ǫ, a typical 2ǫ argument shows

EA(ubu∗) − a1 < 2ǫ. Remark 9. a ≺ b if and only if a + αI ≺ b + αI. Then ΩA(b) = ΩA(b+αI)−αI = EA(UM(b + αI))

sot −αI = EA(UM(b)) sot

So we see that Theorem 8 holds in fact for b ∈ Msa. Corollary 10. For each b ∈ A+, the set EA(UM(b))

sot is convex

and σ-weakly compact.

13

SLIDE 14

Remark 11. The property of EA(UM(b))

sot being convex is equivalent

to Theorem 8. Indeed, if EA(UM(b))

sot is convex,

ΩA(b) = EA(ΩM(b)) = EA

co(UM(b))

sot

⊂ EA (co(UM(b)))

sot

= coEA ((UM(b)))

sot = EA(UM(b)) sot,

while the other inclusion is given by Proposition 1. Arveson and Kadison: if b ∈ Msa, EA

UM(b)
= ΩA(b) ?

Equivalently, EA

UM(b)

sot

= ΩA(b) ?

14

SLIDE 15

Sub-majorization and contractive orbits CM(b) = {vbv∗ : v ∈ M, v ≤ 1}. Θ+

A(b) = {a ∈ A+ : a ≺w b}.

Theorem 12. Let A ⊂ M be a masa and let b ∈ M+. Then EA(CM(b))

sot = Θ+

A(b).

Conjecture 13. Let A ⊂ M be a masa and b ∈ M+. Then EA(CM(b)) = Θ+

A(b).

It turns out that conjecture 13 is equivalent to Arveson-Kadison’s problem: Theorem 14. Let A ⊂ M be a masa. Then the following state- ments are equivalent: (i) EA(UM(b)) = ΩA(b), ∀ b ∈ Msa; (ii) EA(CM(b)) = Θ+

A(b), ∀ b ∈ M+. 15