Fréchet differentiability of the norm of L p -spaces associated with arbitrary von Neumann algebras

Transcription

1 Fréchet differentiability of the norm of L p -spaces associated 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) Fréchet differentiability of L p -norms 1 / 23

2 Definition of Fréchet differentiability The map F between two normed spaces X and Y is called Fréchet differentiable at a point x X if there exists a bounded linear operator F x from X into Y such that for every ɛ > 0 there is δ > 0, for every h X with h X < δ, we have F (x + h) F (x) F x(h) Y ɛ h X. If F is Fréchet differentiable at every point x X, then the derivative F is an operator from X into L(X, Y ) the space of all bounded operators from X into Y. Thus we can consider the second derivative F : 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, Fréchet derivatives are multilinear operators. In the setting of symmetric operator spaces (L p -spaces) examples of such operators are naturally expressed in terms of multiple operator integrals. F. Sukochev (UNSW) Fréchet differentiability of L p -norms 2 / 23

3 Commutative case For 1 p < the question of Fréchet differentiability of the function x x p p on L p 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 been suggested 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 p p, x L p (µ) is infinitely many times Fréchet differentiable whenever p is an even integer; (ii) The function x x p p, x L p (µ) is m-times Fréchet differentiable, where m N is such that m < p m + 1. F. Sukochev (UNSW) Fréchet differentiability of L p -norms 3 / 23

4 The case of l 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 the function f p (x) := x p, x R. Observe that f p C m (R). For k = 1,..., m, x = {x j } j=1, y i = {y (j) i } j=1 l p, i = 1,..., k we denote A (k) x (y 1,..., y k ) = j=1 f p (k) (x j ) y (j) 1... y (j) k. A (k) x defines k-fréchet derivative of x x p p, x l p. F. Sukochev (UNSW) Fréchet differentiability of L p -norms 4 / 23

5 Noncommutative case The question whether the L p -norm of a noncommutative L p -space has the same differentiability properties as the norm of a classical (commutative) L p -space was stated by G. Pisier and Q. Xu in their survey (2003). However, the problem was firstly raised by N. Tomczak-Jaegermann, in 1975 and further emphasized by J. Arazy and Y. Friedman in To resolve this problem in the case of non-commutative L p spaces, the following three natural steps need to be taken: (I) Schatten-von Neumann classes S p ; (II) L p (M, τ) on semifinite von Neumann algebra M, with a faithful normal semifinite trace τ. (III) L p (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)). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 5 / 23

6 Solution of the problem in the case of S p Let B(H) be the space of all bounded linear operators on a Hilbert space H. For 1 p < by S p we denote the closure of the set of all A B(H) such that 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 p p, H S p is infinitely many times Fréchet differentiable, whenever p is an even integer; (ii) The function H H p p, H S p is m-times Fréchet differentiable, where m N is such that m < p m + 1. This theorem confirms the conjecture that the norm of S p and that of their classical counterpart l p share the same differentiability properties; The existence and shape of Fréchet derivatives are based on a new approach to the multiple operator integration theory developed by D. Potapov, A. Skripka, and F. Sukochev, ( Invent. Math. 2013). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 6 / 23

7 The key theorem (Potapov-Sukochev) Theorem If H S p, 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 : S p C, δ (2) H : S p S p C,..., δ (m) H : S p... S p C }{{} m-times such that where V S p. H + V p p H p p m k=1 1 k! δ(k) H (V,..., V ) = O( V p p ), If l p is identified with diagonal operators in S p and x, y l p, then (y,..., y) = A (k) (y,..., y). δ (k) x x F. Sukochev (UNSW) Fréchet differentiability of L p -norms 7 / 23

8 Multiple operator integrals Let A k 1, k 2, be the class of functions φ : R k C admitting the representation k 1 φ(x 0,..., x k 1 ) = a j (x j, ω) dµ(ω), (1) Ω j=0 for some finite measure space (Ω, µ) and bounded Borel functions a j (, ω) : R C. For every φ A k 1, k 2, and a fixed self-adjoint operator H, the multiple operator integral : S p... S p S p T H φ for V j S p, j = 1,..., k 1 is defined as follows Tφ H (V 1,..., V k 1 ) = a 0 (H, ω) V 1 a 1 (H, ω)... V k 1 a k 1 (H, ω) dµ(ω), Ω k 1, where a j s and (Ω, µ) are taken from the representation (1). a simple consequence of the Hölder inequality is that the operator Tφ H is a multilinear bounded operator from S p... S p into S p. k 1 F. Sukochev (UNSW) Fréchet differentiability of L p -norms 8 / 23

9 The divided difference of the zeroth order f [0] is the function f itself. The divided difference of order k = 1,..., m is defined by { f [k 1] f [k] (x 0,x 2,...,x k ) f [k 1] (x 1,x 2,...,x k ) (x 0,..., x k ) := x 0 x 1, if x 0 x 1, d dx 1 f [k 1] (x 1, x 2,..., x k ), if x 0 = x 1, where (x 0,..., x k ) R k+1. Higher derivatives in non-commutative case: the formal explanation d k dt k H + tv p p = d k t=0 dt k Tr( H + tv p ) t=0 ( d k ) ( = Tr dt k H + tv t=0 d k ) p = Tr dt k f p(h + tv ) =..., t=0 where f p is a C compactly supported function on R\{0} such that f p (x) = x p for all x 2 (we assume here that H p, V p 1). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 9 / 23

10 ... A.Skripka = Tr (V d k 1 ) dt k 1 (f p) (H + tv ) t=0 ACDS = (k 1)!Tr ( ) V T H (V,..., V ) (f p ) [k 1] 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] from S p... S p into S p p 1 S p k 1. is bounded Let H = H S p and V S p. 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,..., V ) = (k 1)!Tr T H (V,..., V ), 2 k m. (f p ) [k 1] F. Sukochev (UNSW) Fréchet differentiability of L p -norms 10 / 23

11 τ-measurable operators N is a semifinite von Neumann algebra on the Hilbert space H equipped with faithful normal semifinite trace τ. An (unbounded) operator is said to be affiliated with N if it commutes with every operator in the commutant N of N. Closed densely defined operator A affiliated with N is said to be 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) 1 2 corresponding to the interval (n, ). The collection of all τ-measurable operators is denoted by S(N, τ). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 11 / 23

12 The notions of the distribution function n A, A = A and the singular value function µ(a), A S (N, τ) are defined as follows n A (t) := τ(e A (t, )), t R µ(t; A) := inf{s : n A (s) t}, t 0, where E A (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 a decreasing, right-continuous function on the positive half-line [0, ). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 12 / 23

13 Classical non-commutative L p and L p, The noncommutative space L p (N, τ), 1 p < is defined as follows L p (N, τ) := {A S(N, τ) : µ( A ) L p (0, )}, A p := µ( A ) p, where (L p (0, ), p ) is the usual Lebesgue space. The space L p, (N, τ), 1 p < is the set of all A S(N, τ) such that A p, := sup t 1 p µ(t, A) < +. t 0 If 1 p <, then the space L p, (N, τ) equipped with the quasi-norm p, given above becomes a quasi-banach space. For 1 < p <, there exists a norm p, on L p, (N, τ), which is equivalent to p,. F. Sukochev (UNSW) Fréchet differentiability of L p -norms 13 / 23

14 M is a von Neumann algebra with a faithful normal semifinite weight φ 0. σ φ0 = {σt φ0 } t R is one-parameter modular automorphism group on M N = M σ φ 0 R is a crossed product semifinite von Neumann algebra τ is the canonical semifinite trace on N and θ = {θ s } s R is a dual action For 1 p <, define Haagerup s L p -spaces L p (M) = {A S(N, τ) : θ s (A) = e s p A for all s R} M = {A N : θ s (A) = A for all s R}. There is a linear bijection ψ A ψ between M and L 1 (M). Define the functional tr : L 1 (M) C as tr(a ψ ) := ψ(1), A ψ L 1 (M). tr is the trace on L 1 (M), that is a positive linear functional such that tr(ab) = tr(ba) for all A L 1 (M) and B M. For every A L p (M), 1 p <, it is established that A p L 1 (M). Define a Banach norm on L p (M) by setting A p = tr( A p ) 1 p, A Lp (M). If M is a semifinite von Neumann algebra equipped with a normal faithful semifinite trace τ 0, then the space L p (M) is isometrically isomorphic to the classical non-commutative L p -space L p (M, τ 0 ). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 14 / 23

15 Discussion: Let H = H L p (M) and V L p (M). δ (k) H (V,..., V )? = { A direct approach does not work Tr ( V f p(h) (, ) k = 1 (k 1)!Tr V T H (V,..., V ) (f p ) [k 1], 2 k m For every φ A k 1, fixed self-adjoint operator H, try to define the operator T H φ : L p (M)... L p (M) L p (M), (2) k 1 for V j L p (M), Tφ H (V 1,..., V k 1 ) = j = 1,..., k 1 by Ω a 0 (H, ω) V 1 a 1 (H, ω)... V k 1 a k 1 (H, ω) dµ(ω), where a j 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 Tφ H H L p (M). in L p (M), when k 1 F. Sukochev (UNSW) Fréchet differentiability of L p -norms 15 / 23

16 Indeed, let, for example k = 2 and φ(x 0, x 1 ) = a 0 (x 0 )a 1 (x 1 ), x 0, x 1 R and let a 0, a 1 be bounded Borel functions on R. Taking V L p (M) and H = H M, we have T H φ (V ) = a 0(H)Va 1 (H). Since H M, it follows that a 0 (H), a 1 (H) M. Since L p (M) is a bimodule over M, we have Tφ H is a bounded linear operator. Indeed, : L p(m) L p (M) T H φ (V ) p = a 0 (H)Va 1 (H) p a 0 a 1 V p. F. Sukochev (UNSW) Fréchet differentiability of L p -norms 16 / 23

17 Now take 0 H = H L p (M). Since H S(N, τ), it follows that a 0 (H), a 1 (H) N. a 0 (H), a 1 (H) do not necessarily belong to M. In fact θ s (E H (t, )) = E θs(h)(t, )) = E H (e s p t, ) = E H (t, ), for all s R if and only if t = 0. Thus, E H (t, ) M if and only if t = 0. Thus, for V L p (M), the value T H φ (V ) = a 0(H)Va 1 (H), does not necessarily belong to L p (M). However, in this case, we may definitely say that T H φ (V ) L p, (N, τ) (since L p, (N, τ) is a bimodule over N ). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 17 / 23

18 What does work? For every φ A k 1 and fixed self-adjoint operator H, define the operator T H φ : L p (N, τ)... L p (N, τ) L p (N, τ), k 1 for V j L p (N, τ), j = 1,..., k 1 by Tφ H (V 1,..., V k 1 ) = a 0 (H, ω) V 1 a 1 (H, ω)... V k 1 a k 1 (H, ω) dµ(ω), Ω The operator T H φ is bounded (the Hölder inequality). We extend T H φ to a bounded multilinear operator (denoted again by T H φ ) from L p, (N, τ)... L p, (N, τ) into L p k 1, (N, τ). The crucial step is the fact that the extension T H is a bounded (f p )[k 1] operator from L p, (N, τ)... L p, (N, τ) into L p p 1, (N, τ) (the proof is completely different from that in Potapov-Sukochev). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 18 / 23

19 It follows from the fact µ(t, A) = A p t 1 p, A L p (M), that L p (M) is a closed linear subspace in L p, (N, τ). Thus, the idea is to define δ (k) H on L p, (N, τ)... L p, (N, τ) and then to restrict it to L p (M)... L p (M). Discussion: Let H = H L p (M) and V L p (M). δ (k) H (V,..., V )? = { Tr ( V f p(h) (, ) k = 1 (k 1)!Tr V T H (V,..., V ) (f p )[k 1], 2 k m Take V L p, (N, τ). We have T H (f p )[k 1] (V,..., V ) L p p 1, (N, τ). Thus, V T H (f p )[k 1] (V,..., V ) L 1, (N, τ). We do not request that V T H (f p )[k 1] (V,..., V ) L 1 (M)!!! The key question is What plays the role of Tr on L 1, (N, τ)? F. Sukochev (UNSW) Fréchet differentiability of L p -norms 19 / 23

20 Discussion: Let H = H L p (M) and V L p (M). δ (k) H (V,..., V )? = { Dixmier traces on L 1, (N, τ) Tr ( V f p(h) (, ) k = 1 (k 1)!Tr V T H (V,..., V ) (f p ) [k 1], 2 k m A trace ϕ on L 1, (N, τ) is a linear functional, which is unitarily invariant, i.e. ϕ : L 1, (N, τ) C satisfies ϕ(uau ) = ϕ(a) for all A L 1, (N, τ) and all unitaries U N. A trace ϕ on L 1, (N, τ) is said to be normalized if ϕ(a) = 1, for any 0 A L 1, (N, τ) with µ(t, A) = t 1, t > 0. F. Sukochev (UNSW) Fréchet differentiability of L p -norms 20 / 23

21 Dixmier traces on L 1, (N, τ) For a fixed free ultrafilter ω on N, the functional ϕ : (L 1, (N, τ)) + R +, given by ϕ(a) := lim ω 1 log(1 + n) n 1 µ(t, A)dt,. (3) 0 A L 1, (N, τ), extends to a positive normalized trace on L 1, (N, τ). The key observation in our proof is that for any normalized trace ϕ on L 1, (N, τ) we have that ϕ(a) = tr(a), A L 1 (M), where tr is the trace on L 1 (M). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 21 / 23

22 Definition of the Fréchet derivative For H = H L p, (N, τ), and the trace ϕ on L 1, (N, τ) define the multilinear symmetric functionals δ (k) H,ϕ = δ(k) H on L p, (N, τ)... L p, (N, τ) such that { ϕ(v δ (k) (fp H (V,..., V ) = ) (H)), k = 1 (k 1)!ϕ ( V T H (V,..., V ) ), 2 k m, (f p )[k 1] It was proved by Kosaki that the first Fréchet derivative of H H p p, H L p (M) is given by p tr(v sgn(h) H p 1 ). Observe that ϕ(v (f p ) (H)) = p tr(v sgn(h) H p 1 ). The final crucial step is to show that the restriction of δ (k) H to L p (M)... L p (M) is kth Fréchet derivative of the function H H p p, H L p (M) (see detailed exposition of this proof in the second part of the talk (to be delivered by A. Tomskova)). F. Sukochev (UNSW) Fréchet differentiability of L p -norms 22 / 23

23 The main result Theorem (D. Potapov, F. Sukochev, A. Tomskova, D. Zanin) Let M be a von Neumann algebra equipped with a faithful normal semifinite weight φ 0. Let L p (M) be a non-commutative L p -space, constructed by U. Haagerup. The L p -norm of L p (M) has the same differentiability properties as the norm of a classical commutative L p -space. Corollary Let M be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ. Let L p (M, τ) be a classical non-commutative L p -space. The L p -norm of L p (M, τ) has the same differentiability properties as the norm of a classical commutative L p -space. F. Sukochev (UNSW) Fréchet differentiability of L p -norms 23 / 23