A Schur-Horn theorem in II 1 factors Mart n Argerami and Pedro - PDF document

A Schur-Horn theorem in II 1 factors Mart´ ın Argerami and Pedro Massey Florianopolis, July 2006

Majorization: x, y ∈ R n , k k y ↓ x ↓ � � y ≺ x ⇐ ⇒ j ≤ k = 1 , . . . , n − 1 j j =1 j =1 and n n � � y j = x j . j =1 j =1 Example: if x 1 , . . . , x n ∈ R + , � x i = 1, then (1 n, 1 n, . . . , 1 n ) ≺ ( x 1 , . . . , x n ) ≺ (1 , 0 , . . . , 0) Characterizations: y ≺ x ⇐ ⇒ y ∈ co S n ( x ). ⇐ ⇒ y = Ax , A doubly stochastic. ⇐ ⇒ tr( f ( y )) ≤ tr( f ( x )), for any convex function f . Schur (1923): If A ∈ M n ( C ) sa , then diag( A ) ≺ λ ( A ). 2

A. Horn (1954): If x, y ∈ R n , and y ≺ x , then ∃ A ∈ M n ( C ) sa such that diag( A ) = y, λ ( A ) = x. Schur-Horn (SH): Let x ∈ R n . Then E D ( { UM x U ∗ : U ∈ U n } ) = { M y : 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

Neumann’s answer: in ℓ ∞ ( N ), equality holds if you take closure on both sides: ∞ = { M y : y ≺ x } ∞ . E D ( { UM x U ∗ : U ∈ U n } ) 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 L 1 ( H ) (Arv-Kad 2005): let A ⊂ B ( H ) a discrete masa (i.e. o.n.b.). Let A ∈ L 1 ( H ). Then 1 � � { UAU ∗ : U ∈ U ( H ) } = { B ∈ A ∩ L 1 ( H ) : λ ( B ) ≺ λ ( A ) } . E Related: Kadison’s Carpenter Theorem (what are the diagonals of projections?) 4

How does SH fit in II 1 factors? ( M , τ ). Instead of λ ( x ) ∈ ℓ ∞ ( N ), we have λ x : [0 , 1) → R , λ x ( t ) = min { s ∈ R : τ ( p x ( s, ∞ )) ≤ t } , t ∈ [0 , 1) . (spectral scale/singular numbers: T. Fack, D. Petz, F. Hiai). Properties of λ x : decreasing, right-continuous; � λ x − λ y � ∞ ≤ � x − y � , � λ x − λ y � 1 ≤ � x − y � 1 if x ∈ M + , λ x ( t ) = inf {� xe � : e ∈ P ( M ) , τ (1 − e ) ≤ t } . � 1 τ ( a ) = λ a ( t ) dt. 0 sot , then λ b = λ a Consequence: if b ∈ U M ( a ) 5

Converse: Kamei (1983). So sot = { b ∈ M sa : λ a = λ b } . U M ( a ) = U M ( a ) Three preorders induced by the Spectral Scale Spectral domination: a � b if any of the following holds: (i) λ a ( t ) ≤ λ b ( t ), for all t ∈ [0 , 1). (ii) τ ( p a ( t, ∞ )) ≤ τ ( p b ( t, ∞ )), for all t . Submajorization: a ≺ w b if � s � s λ a ( t ) dt ≤ λ b ( t ) dt, for every s ∈ [0 , 1) . 0 0 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

If N ⊂ M , b ∈ M sa , let Ω N ( b ) = { a ∈ N sa : a ≺ b } , Θ N ( b ) = { a ∈ N sa : a ≺ w b } . Proposition 1. Let N ⊂ M be a von Neumann subalgebra and let E N be the trace preserving conditional expectation onto N . Then, for any b ∈ M sa , (i) E N ( b ) ≺ b . (ii) � E N ( b ) � 1 ≤ � b � 1 . sot ⊂ Ω N ( b ) . (iii) E N ( U M ( b )) sot ⊂ Θ + (iv) If in addition b ∈ M + then E N ( C M ( b )) N ( b ) . 7

Refinements: We say that a ∈ M + has diffuse distribution if s �→ τ ( p a ( 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 p b ( λ, ∞ ) = p a ( 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 h b : [0 , � a ′ � ] → R + 0 such that λ b = λ h b ( a ′ ) . Moreover, a ′ refines b if and only if h b ( a ′ ) = b . (iii) For each c ∈ M + , (a) c � b if and only if h c ≤ h b ; (b) c ≺ w b if and only if h c ( a ′ ) ≺ w h b ( a ′ ) ; (c) c ≺ b if and only if h c ( a ′ ) ≺ h b ( a ′ ) . We say that h b ( a ′ ) is a model of b with respect to a ′ . sot = E A ( U M ( h b ( a ′ ))) sot . Remark: E A ( U M ( b )) 8

Approximations in � · � 1 W ∗ ( a ): the von Neumann subalgebra of M generated by a . Then Ψ a W ∗ ( a ) ≃ L ∞ ( J, ν ) , where ν (∆) = τ ( p a (∆)), σ ( a ) ⊂ J , ϕ ν ( g ) = � g dν , Ψ a : g ( a ) �→ g . Also, λ Ψ a ( g ) = λ g . 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 ) , ϕ ν ) , and let { I i } be an equal-measure partition of [ α, β ] . Let g , h ∈ R 2 n be given by � � g i = 2 n h i = 2 n 1 ≤ i ≤ 2 n . g dν, h dν, I i I i Then g = g ↑ , h = h ↑ , and g ≺ h . 9

Proposition 4. Let g, h : J → R + 0 be increasing left continuous Then g ≺ w h in ( L ∞ ( J, ν ) , ϕ ν ) if and only if there functions. 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 ∈ C M ( b ) = { vbv ∗ : v ∈ M , � v � ≤ 1 } . Definition 6. Let { I ( n ) } 2 n i =1 , n ∈ N , be a ν -dyadic partition of i [ α, β ] . For each n ∈ N and every f ∈ L 1 ( ν ) , x ∈ [ α, β ] , let 2 n � � � � 2 n E n ( f )( x ) = 1 I ( n ) i ( x ) . f dν I ( n ) i =1 i E n is a linear contraction for both � · � 1 and � · � ∞ . 10

Proposition 7. Let { I ( n ) } 2 n i =1 , n ∈ N , be a ν -dyadic partition of i [ α, β ] and let { E n } n ∈ N be the associated family of discrete approx- imations. Then, for every g ∈ L 1 ( ν ) , n →∞ � g − E n ( g ) � 1 = 0 . lim (1) Important idea: diameters in the partition also go to zero, up to measure zero. This makes it work for continuous functions. 11

A Topological Schur-Horn Theorem for II 1 factors Recall: classical SH h ∈ R n . E D ( U n ( M h )) = { M g ∈ D : g ≺ h } , If N ⊂ M , b ∈ M sa , let Ω N ( b ) = { a ∈ N sa : a ≺ b } , Θ N ( b ) = { a ∈ N sa : a ≺ w b } . Theorem 8. Let A ⊂ M be a masa and let b ∈ M + . Then sot = Ω A ( b ) . E A ( U M ( b )) sot ⊃ Ω A ( b ). Proof. By Proposition 1, we only need to prove E A ( U M ( b )) Let a ∈ A + with a ≺ b . ∃ a ′ ∈ A + with diffuse distribution such that it refines a , and sot = E A ( U M ( h b ( a ′ ))) sot . E A ( U M ( b )) So we can assume that b = h b ( a ′ ) ∈ A + . Then h a ( a ′ ) = a and h a ≺ h b in ( L ∞ ( ν ) , ϕ ν ). By Proposition 7, ∃ discrete approximants 2 n 2 n � � � � � � � � � � � a − h i p i � b − g i p i < ǫ, < ǫ � � � � � � � � � � i =1 i =1 1 1 where � � g i = 2 n h i = 2 n 1 ≤ i ≤ 2 n . h a dν, h b dν, I i I i 12

Then, by in Proposition 3, g = g ↑ , h = h ↑ , and g ≺ h . By the classical Schur-Horn theorem there exists U ∈ U n ( C ) such that E D ( UM h U ∗ ) = M g Use U to obtain u , with � 2 n 2 n � � � � � u ∗ h i p i = g i p i E A u i =1 i =1 Since � u ( b − � 2 n i =1 g i p i ) u ∗ � 1 < ǫ, a typical 2 ǫ argument shows � E A ( ubu ∗ ) − a � 1 < 2 ǫ. Remark 9. a ≺ b if and only if a + αI ≺ b + αI . Then sot − αI = E A ( U M ( b )) sot Ω A ( b ) = Ω A ( b + αI ) − αI = E A ( U M ( b + αI )) So we see that Theorem 8 holds in fact for b ∈ M sa . sot is convex Corollary 10. For each b ∈ A + , the set E A ( U M ( b )) and σ -weakly compact. 13

sot being convex is equivalent Remark 11. The property of E A ( U M ( b )) sot is convex, to Theorem 8. Indeed, if E A ( U M ( b )) � sot � sot Ω A ( b ) = E A (Ω M ( b )) = E A co( U M ( b )) ⊂ E A (co( U M ( b ))) sot = E A ( U M ( b )) sot , = co E A (( U M ( b ))) while the other inclusion is given by Proposition 1. if b ∈ M sa , Arveson and Kadison: � � U M ( b ) E A = Ω A ( b ) ? Equivalently, � sot � U M ( b ) E A = Ω A ( b ) ? 14

Sub-majorization and contractive orbits C M ( b ) = { vbv ∗ : v ∈ M , � v � ≤ 1 } . A ( b ) = { a ∈ A + : a ≺ w b } . Θ + Theorem 12. Let A ⊂ M be a masa and let b ∈ M + . Then sot = Θ + E A ( C M ( b )) A ( b ) . Conjecture 13. Let A ⊂ M be a masa and b ∈ M + . Then E A ( C M ( 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) E A ( U M ( b )) = Ω A ( b ) , ∀ b ∈ M sa ; (ii) E A ( C M ( b )) = Θ + A ( b ) , ∀ b ∈ M + . 15