Fréchet differentiability of the norm of Lp-spaces associated with ...

Frechet differentiability of the norm of Lp-spacesassociated with arbitrary von Neumann algebras

(Part I)

Fedor Sukochev(joint work with D. Potapov, A. Tomskova and D. Zanin)

University of NSW, AUSTRALIA

F. Sukochev (UNSW) Frechet differentiability of Lp -norms 1 / 23

Definition of Frechet differentiability

The map F between two normed spaces X and Y is called Frechetdifferentiable at a point x ∈ X if there exists a bounded linearoperator F ′x from X into Y such that for every ε > 0 there is δ > 0,for every h ∈ X with ‖h‖X < δ, we have

‖F (x + h)− F (x)− F ′x(h)‖Y ≤ ε‖h‖X .

If F is Frechet differentiable at every point x ∈ X , then the derivativeF ′ is an operator from X into L(X ,Y ) the space of all boundedoperators from X into Y . Thus we can consider the second derivativeF ′′ : X → L(X , L(X ,Y )), that is F ′′x ∈ L(X , L(X ,Y )) = L(X ×X ,Y ).

The derivative F (k), k ∈ N is defined by induction.

Thus, Frechet derivatives are multilinear operators.

In the setting of symmetric operator spaces (Lp-spaces) examples ofsuch operators are naturally expressed in terms of multiple operatorintegrals.


Commutative case

For 1 ≤ p <∞ the question of Frechet differentiability of thefunction x → ‖x‖pp on Lp space (commutative or non-commutative)has been considered in variety of paper throughout last fifty years.

In the commutative setting the first solution of the problem has beensuggested by R. Bonic and J. Frampton, (J. Math. Mech., 1966)(see also K. Sundaresan, Math. Ann. (1967)).

Theorem (R. Bonic and J. Frampton)

Let (Ω,Σ, µ) be a measurable space with σ-finite measure. Then

(i) The function x → ‖x‖pp, x ∈ Lp(µ) is infinitely many times Frechetdifferentiable whenever p is an even integer;

(ii) The function x → ‖x‖pp, x ∈ Lp(µ) is m-times Frechet differentiable, wherem ∈ N is such that m < p ≤ m + 1.


The case of `p

When p is an even integer, the assertion is trivial

Let 1 < p <∞ and let n ∈ N be such that m < p ≤ m + 1. Consider thefunction fp(x) := |x |p, x ∈ R.

Observe that fp ∈ Cm(R).

For k = 1, ...,m, x = xj∞j=1, yi = y (j)i ∞j=1 ∈ lp, i = 1, ..., k we denote

A(k)x (y1, ..., yk) =

∞∑j=1

f (k)p (xj) y

(j)1 · ... · y

(j)k .

A(k)x defines k-Frechet derivative of x → ‖x‖pp, x ∈ `p.


Noncommutative case

The question whether the Lp-norm of a noncommutative Lp-spacehas the same differentiability properties as the norm of a classical(commutative) Lp-space was stated by G. Pisier and Q. Xu in theirsurvey (2003).

However, the problem was firstly raised by N. Tomczak-Jaegermann,in 1975 and further emphasized by J. Arazy and Y. Friedman in 1992.

To resolve this problem in the case of non-commutative Lp spaces,the following three natural steps need to be taken:

(I) Schatten-von Neumann classes Sp;(II) Lp(M, τ) on semifinite von Neumann algebra M, with a faithful

normal semifinite trace τ.(III) Lp(M) on an arbitrary (type III) von Neumann algebra M.

We answered this question in the affirmative (D. Potapov, F.Sukochev, A. Tomskova, D. Zanin (2014)).


Solution of the problem in the case of Sp

Let B(H) be the space of all bounded linear operators on a Hilbert space H.

For 1 ≤ p <∞ by Sp we denote the closure of the set of all A ∈ B(H) suchthat ‖x‖p := (Tr(x∗x)p/2)1/p <∞, where Tr is the standard trace on B(H).

Theorem (D. Potapov and F. Sukochev, Adv. Math. 2014)

(i) The function H → ‖H‖pp, H ∈ Sp is infinitely many times Frechetdifferentiable, whenever p is an even integer;

(ii) The function H → ‖H‖pp, H ∈ Sp is m-times Frechet differentiable, wherem ∈ N is such that m < p ≤ m + 1.

This theorem confirms the conjecture that the norm of Sp and that of theirclassical counterpart `p share the same differentiability properties;

The existence and shape of Frechet derivatives are based on a new approachto the multiple operator integration theory developed by D. Potapov,A. Skripka, and F. Sukochev, ( Invent. Math. 2013).


The key theorem (Potapov-Sukochev)

Theorem

If H ∈ Sp, 1 < p <∞ and if m ∈ N is such that m < p ≤ m + 1, then

there are bounded symmetric multilinear forms δ(k)H , 1 ≤ k ≤ m

δ(1)H : Sp 7→ C, δ

(2)H : Sp × Sp 7→ C, . . . , δ

(m)H : Sp × . . .× Sp︸︷︷︸

m-times

7→ C

such that

‖H + V ‖pp − ‖H‖pp −

m∑k=1

1

k!δ

(k)H (V , . . . ,V ) = O(‖V ‖pp),

where V ∈ Sp.

If `p is identified with ”diagonal operators” in Sp and x , y ∈ `p, then

δ(k)x (y , ..., y) = A

(k)x (y , ..., y).


Multiple operator integrals

Let Ak−1, k ≥ 2, be the class of functions φ : Rk 7→ C admitting therepresentation

φ(x0, . . . , xk−1) =

∫Ω

k−1∏j=0

aj(xj , ω) dµ(ω), (1)

for some finite measure space (Ω, µ) and bounded Borel functionsaj (·, ω) : R 7→ C.For every φ ∈ Ak−1, k ≥ 2, and a fixed self-adjoint operator H, the multipleoperator integral

THφ : Sp × . . .× Sp 7→ S p

k−1,

for Vj ∈ Sp, j = 1, . . . , k − 1 is defined as follows

THφ (V1, . . . ,Vk−1) =

∫Ω

a0(H, ω) V1 a1(H, ω) · . . . · Vk−1ak−1(H, ω) dµ(ω),

where aj ’s and (Ω, µ) are taken from the representation (1).

a simple consequence of the Holder inequality is that the operator THφ is a

multilinear bounded operator from Sp × . . .× Sp into S pk−1

.F. Sukochev (UNSW) Frechet differentiability of Lp -norms 8 / 23

The divided difference of the zeroth order f [0] is the function f itself. Thedivided difference of order k = 1, . . . ,m is defined by

f [k] (x0, ..., xk) :=

f [k−1](x0,x2,...,xk )−f [k−1](x1,x2,...,xk )

x0−x1, if x0 6= x1,

ddx1

f [k−1](x1, x2, ..., xk), if x0 = x1,

where (x0, ..., xk) ∈ Rk+1.

Higher derivatives in non-commutative case: the formal explanation

dk

dtk‖H + tV ‖pp

∣∣∣t=0

=dk

dtkTr(|H + tV |p)

∣∣∣t=0

= Tr( dk

dtk|H + tV |p

∣∣∣t=0

)= Tr

( dk

dtkfp(H + tV )

∣∣∣t=0

)= ...,

where fp is a C∞ compactly supported function on R\0 such thatfp(x) = |x |p for all |x | ≤ 2 (we assume here that ‖H‖p, ‖V ‖p ≤ 1).


...A.Skripka

= Tr(

V · dk−1

dtk−1(fp)′(H + tV )

∣∣∣t=0

)ACDS

= (k − 1)!Tr(

V · TH(f ′p )[k−1](V , ...,V )

)

We shall consider the operator integral T(f ′p )[k−1] .

The crucial moment is to show that the operator T(f ′p )[k−1] is bounded

from Sp × . . .× Sp into S pp−1⊂ S p

k−1.

Let H = H∗ ∈ Sp and V ∈ Sp. It is known (see e.g. Kosaki, 1984)

that δ(1)H (V ) = pTr

(V sgn(H)|H|p−1

).

Let now 2 < p <∞ and m ∈ N be such that m < p ≤ m + 1. Define

δ(k)H (V , . . . ,V ) = (k − 1)!Tr

(V · TH

(f ′p )[k−1](V , . . . ,V )), 2 ≤ k ≤ m.


τ -measurable operators

N is a semifinite von Neumann algebra on the Hilbert space Hequipped with faithful normal semifinite trace τ .

An (unbounded) operator is said to be affiliated with N if itcommutes with every operator in the commutant N ′ of N .

Closed densely defined operator A affiliated with N is said tobe τ -measurable τ(E|A|(n,∞))→ 0 as n→∞, where E|A|(n,∞) is

the spectral projection of the self-adjoint operator |A| = (A∗A)12

corresponding to the interval (n,∞).

The collection of all τ -measurable operators is denoted by S(N , τ).


The notions of the distribution function nA, A = A∗ and the singular valuefunction µ(A), A ∈ S (N , τ) are defined as follows

nA(t) := τ(EA(t,∞)), t ∈ Rµ(t; A) := infs : n|A|(s) ≤ t, t ≥ 0,

where EA(t,∞) is the spectral projection of the self-adjoint operator A

corresponding to the interval (t,∞).

It follows directly that the singular value function µ(A) is adecreasing, right-continuous function on the positive half-line [0,∞).


Classical non-commutative Lp and Lp,∞

The noncommutative space Lp(N , τ), 1 ≤ p <∞ is defined as follows

Lp(N , τ) := A ∈ S(N , τ) : µ(|A|) ∈ Lp(0,∞), ‖A‖p := ‖µ(|A|)‖p,

where (Lp(0,∞), ‖ · ‖p) is the usual Lebesgue space.

The space Lp,∞(N , τ), 1 ≤ p <∞ is the set of all A ∈ S(N , τ) suchthat

‖A‖′p,∞ := supt≥0

t1pµ(t,A) < +∞.

If 1 ≤ p <∞, then the space Lp,∞(N , τ) equipped with thequasi-norm ‖ · ‖′p,∞ given above becomes a quasi-Banach space. For1 < p <∞, there exists a norm ‖ · ‖p,∞ on Lp,∞(N , τ), which isequivalent to ‖ · ‖′p,∞.


M is a von Neumann algebra with a faithful normal semifinite weight φ0.

σφ0 = σφ0t t∈R is one-parameter modular automorphism group on M

N =Moσφ0 R is a crossed product semifinite von Neumann algebra

τ is the canonical semifinite trace on N and θ = θss∈R is a dual action

For 1 ≤ p <∞, define Haagerup’s Lp-spaces

Lp(M) = A ∈ S(N , τ) : θs(A) = e−sp A for all s ∈ R

M = A ∈ N : θs(A) = A for all s ∈ R.

There is a linear bijection ψ 7→ Aψ between M∗ and L1(M).

Define the functional tr : L1(M)→ C as tr(Aψ) := ψ(1), Aψ ∈ L1(M).

tr is the trace on L1(M), that is a positive linear functional such thattr(AB) = tr(BA) for all A ∈ L1(M) and B ∈M.

For every A ∈ Lp(M), 1 ≤ p <∞, it is established that |A|p ∈ L1(M).

Define a Banach norm on Lp(M) by setting ‖A‖p = tr(|A|p)1p , A ∈ Lp(M).

If M is a semifinite von Neumann algebra equipped with a normal faithfulsemifinite trace τ0, then the space Lp(M) is isometrically isomorphic to theclassical non-commutative Lp-space Lp(M, τ0).


Discussion: Let H = H∗ ∈ Lp(M) and V ∈ Lp(M).

δ(k)H (V , . . . ,V )

?=

Tr(V f ′p(H)

), k = 1

(k − 1)!Tr(

V · TH(f ′p )[k−1](V , . . . ,V )

), 2 ≤ k ≤ m

A direct approach does not work

For every φ ∈ Ak−1, fixed self-adjoint operator H, try to define the operator

THφ : Lp(M)× . . .× Lp(M) 7→ L p

k−1(M), (2)

for Vj ∈ Lp(M), j = 1, . . . , k − 1 by

THφ (V1, . . . ,Vk−1) =

∫Ω


where aj ’s and (Ω, µ) are taken from the representation (1).

The definition above makes sense in case H ∈M (next slide).

It is NOT clear whether the values of THφ in L p

k−1(M), when

H ∈ Lp(M).F. Sukochev (UNSW) Frechet differentiability of Lp -norms 15 / 23

Indeed, let, for example k = 2 and φ(x0, x1) = a0(x0)a1(x1),x0, x1 ∈ R and let a0, a1 be bounded Borel functions on R.Taking V ∈ Lp(M) and H = H∗ ∈M, we have

THφ (V ) = a0(H)Va1(H).

Since H ∈M, it follows that a0(H), a1(H) ∈M.Since Lp(M) is a bimodule over M, we have TH

φ : Lp(M)→ Lp(M)is a bounded linear operator. Indeed,

‖THφ (V )‖p = ‖a0(H)Va1(H)‖p ≤ ‖a0‖∞‖a1‖∞‖V ‖p.


Now take 0 6= H = H∗ ∈ Lp(M).

Since H ∈ S(N , τ), it follows that a0(H), a1(H)∈ N .a0(H), a1(H) do not necessarily belong to M. In fact

θs(EH(t,∞)) = Eθs(H)(t,∞)) = EH(esp t,∞) = EH(t,∞),

for all s ∈ R if and only if t = 0.

Thus, EH(t,∞) ∈M if and only if t = 0.

Thus, for V ∈ Lp(M), the value THφ (V ) = a0(H)Va1(H), does not

necessarily belong to Lp(M).

However, in this case, we may definitely say thatTHφ (V ) ∈ Lp,∞(N , τ) (since Lp,∞(N , τ) is a bimodule over N ).


What does work?

For every φ ∈ Ak−1 and fixed self-adjoint operator H, define the operator

THφ : Lp(N , τ)× . . .× Lp(N , τ) 7→ L p

k−1(N , τ),

for Vj ∈ Lp(N , τ), j = 1, . . . , k − 1 by

THφ (V1, . . . ,Vk−1) =

∫Ω


The operator THφ is bounded (the Holder inequality).

We extend THφ to a bounded multilinear operator (denoted again by TH

φ )from Lp,∞(N , τ)× . . .× Lp,∞(N , τ) into L p

k−1 ,∞(N , τ).

The crucial step is the fact that the extension TH(f ′p )[k−1] is a bounded

operator from Lp,∞(N , τ)× . . .× Lp,∞(N , τ) into L pp−1 ,∞(N , τ) (the proof

is completely different from that in Potapov-Sukochev).


It follows from the fact µ(t,A) = ‖A‖pt−1p , A ∈ Lp(M), that

Lp(M) is a closed linear subspace in Lp,∞(N , τ).

Thus, the idea is to define δ(k)H on Lp,∞(N , τ)× . . .× Lp,∞(N , τ) and

then to restrict it to Lp(M)× . . .× Lp(M).


δ(k)H (V , . . . ,V )

?=

Tr(V f ′p(H)

), k = 1

(k − 1)!Tr(

V · TH(f ′p )[k−1](V , . . . ,V )

), 2 ≤ k ≤ m

Take V ∈ Lp,∞(N , τ). We have TH(f ′p )[k−1](V , . . . ,V ) ∈ L p

p−1,∞(N , τ).

Thus, V · TH(f ′p )[k−1](V , . . . ,V ) ∈ L1,∞(N , τ).

We do not request that V · TH(f ′p )[k−1](V , . . . ,V ) ∈ L1(M)!!!

The key question is ”What plays the role of Tr on L1,∞(N , τ)?”



δ(k)H (V , . . . ,V )

?=

Tr(V f ′p(H)

), k = 1

(k − 1)!Tr(

V · TH(f ′p )[k−1](V , . . . ,V )

), 2 ≤ k ≤ m

Dixmier traces on L1,∞(N , τ)

A trace ϕ on L1,∞(N , τ) is a linear functional, which is unitarilyinvariant, i.e. ϕ : L1,∞(N , τ)→ C satisfies ϕ(UAU∗) = ϕ(A) for allA ∈ L1,∞(N , τ) and all unitaries U ∈ N .

A trace ϕ on L1,∞(N , τ) is said to be normalized if ϕ(A) = 1, for any0 ≤ A ∈ L1,∞(N , τ) with µ(t,A) = t−1, t > 0.


Dixmier traces on L1,∞(N , τ)

For a fixed free ultrafilter ω on N, the functionalϕ : (L1,∞(N , τ))+ → R+, given by

ϕ(A) := limω

1

log(1 + n)

∫ n

1µ(t,A)dt, . (3)

0 ≤ A ∈ L1,∞(N , τ), extends to a positive normalized trace onL1,∞(N , τ).

The key observation in our proof is that for any normalized trace ϕ onL1,∞(N , τ) we have that

ϕ(A) = tr(A), A ∈ L1(M),

where tr is the trace on L1(M).


Definition of the Frechet derivative

For H = H∗ ∈ Lp,∞(N , τ), and the trace ϕ on L1,∞(N , τ) define the

multilinear symmetric functionals δ(k)H,ϕ = δ

(k)H on

Lp,∞(N , τ)× . . .× Lp,∞(N , τ) such that

δ(k)H (V , . . . ,V ) =

ϕ(V · (fp)′(H)), k = 1(k − 1)!ϕ

(V · TH

(f ′p )[k−1](V , . . . ,V )), 2 ≤ k ≤ m, .

It was proved by Kosaki that the first Frechet derivative ofH 7→ ‖H‖pp, H ∈ Lp(M) is given by p tr(V · sgn(H)|H|p−1).

Observe that ϕ(V · (fp)′(H)) = p tr(V · sgn(H)|H|p−1).

The final crucial step is to show that the restriction of δ(k)H to

Lp(M)× . . .× Lp(M) is kth Frechet derivative of the functionH → ‖H‖pp, H ∈ Lp(M) (see detailed exposition of this proof in thesecond part of the talk (to be delivered by A. Tomskova)).


The main result

Theorem (D. Potapov, F. Sukochev, A. Tomskova, D. Zanin)

Let M be a von Neumann algebra equipped with a faithful normalsemifinite weight φ0. Let Lp(M) be a non-commutative Lp-space,constructed by U. Haagerup.The Lp-norm of Lp(M) has the same differentiability properties as thenorm of a classical commutative Lp-space.

Corollary

Let M be a semifinite von Neumann algebra equipped with a faithfulnormal semifinite trace τ. Let Lp(M, τ) be a classical non-commutativeLp-space.The Lp-norm of Lp(M, τ) has the same differentiability properties as thenorm of a classical commutative Lp-space.


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