Decomposable Schur multipliers and non-commutative Fourier - - PowerPoint PPT Presentation

Decomposable Schur multipliers and non-commutative Fourier multipliers

Christoph Kriegler (Clermont-Ferrand), joint work with C´ edric Arhancet (Besan¸ con) Jussieu – 20 October 2015

Regular operators on classical Lp(Ω) spaces

Let 1 ≤ p ≤ ∞, and (Ωk, µk) be two σ-finite measure spaces (k = 1, 2). An operator T : Lp(Ω1) → Lp(Ω2) is called positive if for f ∈ Lp(Ω1), f ≥ 0 pointwise, we always have Tf ≥ 0 pointwise. An operator T : Lp(Ω1) → Lp(Ω2) is called regular if T = T1 − T2 + i(T3 − T4) with T1, T2, T3, T4 positive operators. THEOREM: Let T : Lp(Ω1) → Lp(Ω2) be a regular operator, X a Banach space and S : X → X a bounded operator. Then the tensor product T ⊗ S : Lp(Ω1) ⊗ X ⊂ Lp(Ω1; X) → Lp(Ω2; X) extends to a bounded operator on the Bochner space Lp(Ω1; X) with T ⊗ S ≤ Treg S. Here, Treg = supn∈N T ⊗ Iℓ∞

n Lp(Ω1;ℓ∞ n )→Lp(Ω2;ℓ∞ n ) < ∞.

Schatten classes and non-commutative Lp(M) spaces

Let I be a non-empty index set and 1 ≤ p < ∞. Then the Schatten class Sp

I is defined to be the class of all compact

• perators T on ℓ2

I such that tr((T ∗T)p/2) < ∞.

S∞

I

= {compact operators on ℓ2

I }.

Let M ⊂ B(H) be a von Neumann algebra, i.e. weak∗ closed involutive subalgebra of B(H). Assume that M is equipped with a semifinite faithful normal trace τ : M+ → [0, ∞]. Then for 1 ≤ p < ∞, the non-commutative Lp space is defined to be: Lp(M) = Lp(M, τ) = completion of {x ∈ M : xLp(M) = τ((x∗x)p/2)

1 p < ∞}.

L∞(M) := M. For example, Lp(Ω) = Lp(L∞(Ω),

• Ω ·dµ), and Sp

I = Lp(B(ℓ2 I ), tr)

for 1 ≤ p < ∞.

Completely bounded and completely positive mappings

Sp

I and Lp(M) are Banach spaces, but even more:

Let n ∈ N. Define a norm on Mn ⊗ Lp(M) = Mn(Lp(M)) by [Pisier] [xij]Mn(Lp(M)) = sup{α · x · βLp(Mn(M)) : αS2p

n , βS2p n ≤ 1}.

Lp(M) is called an operator space equipped with the sequence of norms on Mn(Lp(M)), n ∈ N. A mapping u : Lp(M1) → Lp(M2) is called completely bounded if the family of mappings un : Mn(Lp(M1)) → Mn(Lp(M2)), [xij] → [u(xij)] satisfy ucb = supn∈N un < ∞. Further, u is called completely positive, if all the mappings un are positive, where x ∈ Mn(Lp(M1)) is defined to be positive if x = y∗y with y ∈ Mn(L2p(M1)).

Completely positive mappings and classical Lp(Ω) spaces

PROPOSITION: Let 1 ≤ p ≤ ∞. Let Lp(Ω) be a classical Lp space and Lp(M) a non-commutative one. Then a positive mapping u : Lp(M) → Lp(Ω) is completely positive. Idea of proof: Let a ∈ Mn,1 and [xij] ∈ Mn(Lp(M)) positive. Then a∗[xij]a ∈ Lp(M) is positive, hence u(a∗[xij]a) ∈ Lp(Ω) is positive. Thus, a∗[u(xij)(ω)]a =

n

• i,j=1

aiu(xij)(ω)aj = u(

n

• i,j=1

aixijaj)(ω) = u(a∗[xij]a)(ω) ≥ 0. Then for a.a. ω ∈ Ω, [u(xij)(ω)] is positive in Mn. Now use the fact that [fij] ∈ Mn(Lp(Ω)) is positive if and only if [fij(ω)] ∈ Mn is a positive matrix for almost all ω ∈ Ω. Hence [u(xij)] is positive in Mn(Lp(Ω)). COROLLARY: Any positive mapping u : Lp(Ω) → Lp(M) is completely positive.

Definition of decomposable mappings

DEFINTION: Let 1 ≤ p ≤ ∞ and T : Lp(M1) → Lp(M2) be a bounded linear mapping. Then T is called decomposable if T = T1 − T2 + i(T3 − T4) with completely positive mappings T1, T2, T3, T4. The set of decomposable operators Dec(Lp(M1), Lp(M2)) is a Banach space equipped with the norm Td = sup

|λ|≤1

inf{T1 + T2 + T3 + T4 : λT = T1 − T2 + i(T3 − T4)}.

Equivalent norm for decomposable mappings

PROPOSITION: Let 1 ≤ p ≤ ∞ and let T : Lp(M1) → Lp(M2) be a bounded linear mapping. Then T is decomposable if and only if there exist v1, v2 : Lp(M1) → Lp(M2) such that the mapping v1 T T ◦ v2

• : Sp

2 (Lp(M1)) → Sp 2 (Lp(M2))

is completely positive, where T ◦(x) = (T(x∗))∗. We let Tdec = inf {max{v1cb, v2cb}} , where the infimum runs

• ver all possible v1, v2. Then Td and Tdec are equivalent on

Dec(Lp(M1), Lp(M2)).

Properties of decomposable mappings

PROPOSITION: Let M1, M2 be QWEP von Neumann algebras. Let 1 < p < ∞. Then any decomposable map T : Lp(M1) → Lp(M2) is completely bounded and Tcb ≤ Tdec. In particular, any completely positive mapping T : Lp(M1) → Lp(M2) is completely bounded. THEOREM [Pisier]: Let M1, M2 be hyperfinite von Neumann

• algebras. Then T : Lp(M1) → Lp(M2) is decomposable if and only

if for any operator space E, T ⊗ IE : Lp(M1; E) → Lp(M2; E) is

• bounded. In this case, in fact T ⊗ IE ≤ CTdec < ∞,

and supn∈N T ⊗ IMnLp(M1;Mn)→Lp(M2;Mn) ∼ = Td ∼ = Tdec.

Decomposable vs. completely bounded mappings

PROPOSITION [Haagerup p = ∞, A.-K.]: Let M have a finite trace τ and u1, . . . , un ∈ M be arbitrary unitaries. Let 1 ≤ p ≤ ∞. Consider the map T : ℓp

n → Lp(M) defined by T(ek) = uk. Then

Tdec = n1− 1

p .

Consider now Fn the free group of n generators g1, g2, . . . , gn, and VN(Fn) the group von Neumann algebra, contained in B(ℓ2(Fn)), generated by the unitary elements λs(f ) = f (s−1·). THEOREM [Haagerup p = ∞, A.-K.]: Let 1 ≤ p ≤ ∞. Let n ≥ 2 be an integer. The map Tn : ℓp

n → Lp(VN(Fn)), ek → λgk satisfies

Tncb ≤ (2√n − 1)1− 1

p and Tndec = n1− 1 p . In particular,

Tndec/Tncb → ∞ as n → ∞.

Open questions

Question 1: Let R be the hyperfinite factor of type II1 and let U1, . . . , Un ∈ R be a sequence of self-adjoint anticommuting

• perators. Suppose 1 ≤ p ≤ ∞. Consider the map T : ℓp

n → Lp(R)

defined by T(ek) = Uk. What are the values of T, Tdec, Tcb ? Question 2: Let 1 ≤ p ≤ ∞. Do we have for every map T : ℓp

2 → Lp(M) the equalities T = Tcb = Tdec ?

True for p = ∞ [Haagerup]. Question 3: Let 1 ≤ p ≤ ∞. Suppose that for every map T : ℓp

3 → Lp(M) we have T = Tcb = Tdec.

Is M necessarily hyperfinite? Even open for p = ∞.

Definition of Schur multipliers

Let I be some index set, 1 ≤ p ≤ ∞, and φ : I × I → C be a bounded function. A mapping Mφ : Sp

I → Sp I is called Sp-Schur

multiplier if it is of the form Mφ([xij]) = [φ(i, j)xij].

Complementation of Schur multipliers

THEOREM [A.-K.]: Let I be some index set. For a completely bounded mapping S : Sp

I → Sp I let

φS : I × I → C, (i, j) → tr(S(eij)eji). Then the linear mapping PI : CB(Sp

I ) → CB(Sp I ), S → MφS

has the following properties:

• 1. PI takes its values in the completely bounded Sp-Schur

multipliers.

• 2. PI is contractive.
• 3. PI(S) = S as soon as S is already a cb Sp-Schur multiplier.
• 4. PI(S) is completely positive as soon as S is completely

positive.

Proof of Complementation of Schur multipliers

PROOF: Let ∆ : B(ℓ2

I ) → B(ℓ2 I )⊗B(ℓ2 I ) be the normal

∗-isomorphism which preserves the traces onto the sub von Neumann algebra ∆(B(ℓ2

I )) ⊆ B(ℓ2 I )⊗B(ℓ2 I ) such that

∆(eij) = eij ⊗ eij, (i, j ∈ I). Let E be the normal conditional expectation of B(ℓ2

I )⊗B(ℓ2 I ) onto

∆(B(ℓ2

I )) that leaves tr ⊗ tr invariant. For any i, j, k, l ∈ I we have

E(eij ⊗ ekl) = δikδjleij ⊗ eij. Set now PI(S) = ∆−1E(S ⊗ IdSp

I )∆.

If S completely positive, then also PI(S) is. Moreover, PI(S)cb,Sp

I →Sp I ≤ ∆−1E(S ⊗ IdSp I )∆cb

≤ Scb,Sp

I →Sp I .

Finally check that PI(S) is a Schur multiplier and PI(S) = S if S is already a Schur multiplier.

Consequences of the complementation

COROLLARY: Let I be an index set, 1 < p < ∞ and φ : I × I → C a bounded function. Then Mφ is a decomposable Sp-Schur multiplier if and only if Mφ is a bounded Schur multiplier B(ℓ2

I ) → B(ℓ2 I ).

Proof: “= ⇒”: Let Mφ : Sp

I → Sp I be decomposable. Then

Mφ = R1 − R2 + i(R3 − R4) with completely positive Rk. Thus, Mφ = PI(Mφ) = PI(R1) − PI(R2) + i(PI(R3) − PI(R4)). Now each PI(Rk) is a completely positive Sp-Schur multiplier, which is known to be bounded on B(ℓ2

I ). Thus, also Mφ = PI(Mφ) is

bounded on B(ℓ2

I ).

“⇐ =”: Mφ bounded on B(ℓ2

I ) =

⇒ completely bounded on B(ℓ2

I ) =

⇒ decomposable on B(ℓ2

I ) =

⇒ decomposable on Sp

I .

Strongly non decomposable operators

DEFINITION ([Arendt-Voigt 1991] in the case of Fourier multipliers on abelian groups): Let T : Lp(M1) → Lp(M2) be completely bounded. T is called CB strongly non decomposable if T does not belong to Dec(Lp(M1), Lp(M2)), closure in CB(Lp(M1), Lp(M2)).

Existence of CB strongly non decomposable Schur multipliers

PROPOSTION [A.-K.]: Let I be an index set. Suppose 1 < p < ∞. Let φ : I × I → C be bounded. If Mφ : Sp

I → Sp I

belongs to the closure Dec(Sp

I ) in CB(Sp I ), then φ belongs to the

closure of {“B(ℓ2

I )-Schur mult. functions” } in ℓ∞(I × I).

Proof: uses again complementation of Sp-Schur multipliers. COROLLARY: The triangular truncation Sp

Z → Sp Z, [xij] → [δi≤jxij]

is CB strongly non decomposable. Proof: By contraposition, use the known fact [Bennett 1977] that a bounded Schur multiplier Mφ on B(ℓ2

Z) with existing limits

limi limj φij and limj limi φij has equal limits.

Continuous Schur multipliers

DEFINITION: Let (Ω, µ) be a σ-finite measure space and φ : Ω × Ω → C a measurable function. Let Sp

Ω = Lp(B(L2(Ω, µ)), tr). A mapping Mφ : Sp Ω → Sp Ω is called

(continuous) Schur multiplier if it maps an element x ∈ S2

Ω ∩ Sp Ω

identified with a function (ω1, ω2) → x(ω1, ω2) in L2(Ω × Ω), to (ω1, ω2) → φ(ω1, ω2)x(ω1, ω2). PROPOSITION: There exists a linear mapping PΩ : CB(Sp

I ) → CB(Sp I )

with the following properties:

• 1. PΩ takes its values in the completely bounded Sp

Ω-Schur

multipliers.

• 2. PΩ is contractive.
• 3. PΩ(S) = S as soon as S is already a cb Sp-Schur multiplier.
• 4. PΩ(S) is completely positive as soon as S is completely

positive.

Proof Complementation Continuous Schur multipliers

ELEMENTS OF THE PROOF: Let A1, . . . , An be mutually disjoint measurable subsets of Ω. Let En : L2(Ω) → L2(Ω), f → n

k=1 1 µ(Ak)f , 1Ak1Ak be the

associated conditional expectation. Then we can use the complementation of discrete Schur multipliers PIn(ΨnSΦn) with Ψn : Sp

Ω → Sp In, x → EnxE ′ n and Φn : Sp In → Sp Ω, x → E ′

• nxEn. Use

now a w∗ accumulation argument to capture a limit point PΩ(S) = lim ΦnPIn(ΨnSΦn)Ψn.

Definition of Fourier multipliers I

Goal: define non-commutative Fourier multipliers. Let m : R → C be a bounded measurable function. An Lp-Fourier multiplier on R is a mapping of the form Tmf = F−1[mˆ f ] =

• R

m(s)ˆ f (s)eis(·)ds which extends boundedly to Lp(R) → Lp(R). For s ∈ R, consider χs : L2(R) → L2(R), f → eis(·)f (·), which is a unitary mapping. We have χs1χs2 = χs1+s2, so χ : R → B(L2(R)), s → χs is a group homomorphism with values in the unitaries.

Definition of Fourier multipliers II

Now replace in the above R by a locally compact group G (not necessarily abelian), equipped with left Haar measure. We put for s ∈ G λs : L2(G) → L2(G), f → f (s−1·). Then λs1λs2 = λs1s2, so that λ : G → B(L2(G)) is a

• homomorphism. We set M = VN(G) the von Neumann algebra

generated by {λs : s ∈ G}. Let now m : G → C be a bounded measurable function. VN(G) is equipped with the functional τ(λg) = δge, which extends to a trace if G is unimodular. For f belonging to a dense subset of Lp(VN(G)), we can write f =

• G ˆ

f (s)λsds for some bounded measurable function ˆ f : G → C. An Lp-Fourier multiplier on G is a mapping of the form Tmf =

• G

m(s)ˆ f (s)λsds which extends to a bounded operator on Lp(VN(G)).

Complementation of Fourier multipliers

THEOREM: Let G be a discrete group. For a completely bounded mapping S : Lp(VN(G)) → Lp(VN(G)) let mS : G → C, s → τ(S(λs)λ∗

s). Then the linear mapping

PG : CB(Lp(VN(G))) → CB(Lp(VN(G))), S → TmS has the following properties:

• 1. PG takes its values in the completely bounded Lp-Fourier

multipliers.

• 2. PG is contractive.
• 3. PG(S) = S as soon as S is already a cb Lp-Fourier multiplier.
• 4. PG(S) is completely positive as soon as S is completely

positive.

Application: Strongly non decomposable Fourier multipliers

QUESTION: Given a locally compact group G and 1 < p < ∞, does there exist a CB strongly non decomposable Fourier multiplier

• n Lp(VN(G))?

PROPOSITION [Arendt-Voigt 1991]: Let G be an abelian group. If Tm : Lp(VN(G)) → Lp(VN(G)) belongs to Dec(Lp(VN(G))), then m : G → C is continuous. EXAMPLE [Arendt-Voigt 1991]: Let G = R. Then the Fourier multiplier Tm with symbol m(t) = sign(t) is CB strongly non decomposable.

Strongly non decomposable Fourier multipliers

PROPOSITION [A.-K.]: Let G be a discrete group and m : G → C a bounded measurable function.

• 1. Let 1 ≤ p ≤ ∞. If Tm ∈ Dec(Lp(VN(G)))

CB(Lp(VN(G))), then

m belongs to the closure of the completely bounded L∞-Fourier multipliers, closure in ℓ∞

G .

• 2. If limn m(gn) and limn m(g−n) exist for some g ∈ G, and m

belongs to the above closure, then necessarily limn m(gn) = limn m(g−n).

• 3. Let 1 < p < ∞, n ∈ N and G = Fn the free group of n
• generators. Then there exists a CB strongly non

decomposable self-adjoint Fourier multiplier on Lp(VN(Fn)).

Proof of Proposition

PROOF: 1. Show that Tm − RLp(VN(G))→Lp(VN(G)) ≥ distℓ∞

G (m, cb L∞-mult.) for any

R ∈ Dec(Lp(VN(G))) : We can write R = R1 − R2 + i(R3 − R4) with Rj completely positive. By the preceding Theorem, PG(Rj) is a completely positive Fourier multiplier, so PG(R) is a Fourier multiplier belonging to Dec(Lp(VN(G))). = ⇒ PG(R) ∈ Dec(VN(G)) ⊆ CB(VN(G)). We deduce Tm − RCB(Lp(VN(G))) ≥ PG(Tm) − PG(R)cb,p→p ≥ Tm − PG(R)2→2 ≥ dℓ∞

G (Tm, cb L∞-mult.)

Proof of Proposition

• 2. Assume first Tm is a cb. L∞ Fourier multiplier.

Then according to [Bozejko Fendler 1984] there exist a Hilbert space H and P, Q : G → H with supx∈G P(x)H, Q(x)H < ∞ such that m(y−1x) = P(x), Q(y)H for x, y ∈ G. (P(gi))i and (Q(gj))j are bounded sequences in H, thus admit w∗ convergent

• subsequences. Then it follows that

limi→∞ m(gi−j) = limi→∞ m(gi) = limi→∞P(gi), Q(gj) = limk P(gik), limk P(gjk). Similarly, limj→∞ m(gi−j) = limk P(gik), limk Q(gjk). Thus, limn m(gn) = limn m(g−n). If Tm belongs only to the ℓ∞

G closure of the cb. L∞ Fourier

multipliers, use a simple approximation argument.

Proof of Proposition

• 3. We can choose the Fourier multiplier on Lp(VN(Fn)) a

non-commutative Riesz transform from [Junge Mei Parcet 2014]: The symbol is m(g) = b(g), hH/

• ψ(g) for some representing

real Hilbert space H, a “length function” ψ : G → R+, and an affine representation b : G → H, b(gj1

i1 . . . gjN iN ) = j1hi1 +. . .+jNhiN.

Since m is real valued, Tm is self-adjoint. Moreover, m(gn

1 ) = sign(n), so that limn m(gn 1 ) = 1 = −1 = limn m(g−n 1 ).

Thus, by parts 1. and 2., Tm is CB strongly non decomposable.

Complementation of Fourier multipliers for non-discrete groups

The complementation of Fourier multipliers PG : CB(Lp(VN(G))) → CB(Lp(VN(G))) assumed that G is

• discrete. In fact, if G is discrete, the trace τ(λg) = δge is finite, so

that the unitaries λg generating VN(G) belong to Lp(VN(G)) and ∆ : VN(G) → VN(G)⊗ VN(G), λg → λg ⊗ λg extends naturally to a bounded operator on Lp(VN(G)). This breaks down if G is not discrete and p = ∞. But: some non-discrete groups admit an approximation by discrete groups. DEFINITION: [Caspers Parcet Perrin Ricard 2014]: Let G be a locally compact group. G is called ADS (approximable by discrete subgroups) if there exists a family of lattices (Γj)j≥1 in G and associated fundamental domains (Xj)j≥1 which form a neighborhood basis of the identity. In this case, G is unimodular.

Fourier multiplier complementation for ADS groups

THEOREM [A.-K.]: Let G be an amenable ADS group. Assume that the fundamental domains are symmetric, i.e. µ(X −1

j

∆Xj) = 0, where µ is left Haar measure. Assume moreover that 1 µ(Xj)

• G

µ(Xj ∩ γXjg)2 µ(Xj)2 dµ(g) → c (j → ∞) (1) for some c > 0, uniformly in γ ∈ Γj. Then for 1 ≤ p ≤ ∞ there exists a linear mapping PG : CB(Lp(VN(G))) → CB(Lp(VN(G)))

• f norm at most 1

c with the properties:

• 1. PG(T) is a Fourier multiplier.
• 2. If T is completely positive, then PG(T) is completely positive.
• 3. If T = Tm is a Fourier multiplier on Lp(VN(G)) with

uniformly continuous symbol m : G → C, then PG(Tm) = Tm.

Fourier multiplier complementation for ADS groups

PARTS OF THE PROOF: Use an approximation idea of [Caspers Parcet Perrin Ricard 2014]: Let hj =

• Xj λsdµ(s) ∈ VN(G) and

define for 1 ≤ p ≤ ∞ : Φp

j : Lp(VN(Γj)) → Lp(VN(G)), λγ → µ(Xj)−2+ 1

p h∗

j λγhj.

Then Φp

j is completely positive and completely contractive for any

1 ≤ p ≤ ∞. Further let Ψp

j =

• Φp∗

j

∗ : Lp(VN(G)) → Lp(VN(Γj)). Define now for given T ∈ CB(Lp(VN(G))) the Fourier multiplier

• n the discrete group Γj

Tmj = 1 c PΓj(Ψp

j TΦp j ).

Let further the mollified Fourier multiplier symbol on G :

• mj =

1 µ(Xj)1Xj ∗ (mjµΓj) ∗ 1Xj. Now show that T mj converges to a

CB Lp-Fourier multiplier on G and use (1) to show that T

mj

converges to Tm if T = Tm.

More on ADS groups

COROLLARY:

• 1. Let G0 be a discrete amenable group and G1 an LCA group

which is ADS, satisfying (1) for some c > 0, for example G1 = Rn with c = ( 2

3)n. Let G0 act on G1 via a suitable

homomorphism φ : G0 → Aut(G1). Then the semidirect product G = G0 ⋉φ G1 is amenable ADS and (1) holds. Consequently, the above Theorem applies.

• 2. Let G1 be an LCA compactly generated Lie group, so

isomorphic to Rn × Tm × Zl × F with F a finite abelian

• group. Then G1 is ADS and satisfies (1). Moreover, for G0 a

subgroup of Sn × Sm there exists a nontrivial homomorphism φ : G0 → Aut(G1) as in 1., exchanging “axes” in G1. Consequently, the Theorem applies. OPEN QUESTION: Find non-abelian Lie groups satisfying (1).

Existence of strongly non regular Fourier multipliers

QUESTION: Let G be a locally compact abelian group and 1 < p < ∞. Does there exist a strongly non regular Fourier multiplier (snrFm) on Lp(G), i.e. a bounded Lp Fourier multiplier not belonging to Dec(Lp(G)) ? Observations:

• 1. If G is finite, then a finite dimension argument shows that no

strongly non regular Fourier multiplier can exist.

• 2. If G = R, Z or T, then by [Arendt-Voigt 1991], the Hilbert

transform is an example of a strongly non regular Fourier multiplier on Lp(G).

• 3. For LCA groups, VN(G) = L∞(ˆ

G), where ˆ G is again an LCA group, the Pontryagin dual.

Structure Theorems of stongly non regular Fourier multipliers

IDEA: Try to pass from a snrFm on a subgroup/quotient group to a snrFm on the whole group. For H ⊆ G, H⊥ = {ξ ∈ ˆ G : ξ, h = 1 for all h ∈ H}. PROPOSITION: Let G be a LCA group and H a compact subgroup of G. If m : H⊥ → C is a complex function, we denote by ˜ m : ˆ G → C the extension of m which is zero off H⊥. If Tm induces a snrFm Tm : Lp(G/H) → Lp(G/H), then ˜ m induces a snrFm T ˜

m : Lp(G) → Lp(G).

PARTS OF THE PROOF: Suppose that T ˜

m belongs to

Dec(Lp(G)). Let ǫ > 0. There exist some positive maps Rj : Lp(G) → Lp(G) and a bounded map R : Lp(G) → Lp(G) of norm < ǫ such that T ˜

m = R1 − R2 + i(R3 − R4) + R. Using

complementation we can assume that Rj and R are Fourier multipliers. Show that Rj and R pass to the quotient group G/H.

Structure Theorems of strongly non regular Fourier multipliers

PROPOSITION: Let G be a LCA group and H be a closed subgroup of G. Denote π : ˆ G → ˆ G/H⊥ the canonical map. Let m : ˆ H → C be a complex function. Then m ◦ π : ˆ G → C induces a snrFm Lp(G) → Lp(G) if and only if m : ˆ H → C induces a snrFm Lp(H) → Lp(H). PROPOSITION: Let G be an infinite compact abelian group. Then there exists a snrFm Lp(G) → Lp(G), 1 < p < ∞. PARTS OF THE PROOF: G compact = ⇒ ˆ G discrete. If ˆ G contains an element of infinite order, then ˆ G ⊇ Z. Use [Arendt-Voigt 1991] and the above structure proposition to find a

• snrFm. Otherwise, ˆ

G is torsion. Use abstract Paley-Littlewood multiplier theory to find a snrFm of the form m = ∞

n=0 1Y2n+1 − 1Y2n, where (Yn)n is an increasing exhaustive

sequence of finite subgroups of ˆ G.

Existence of strongly non regular Fourier multipliers

PROPOSITION: Let G be an infinite discrete abelian group. Then there exists a strongly non regular Fourier multiplier on Lp(G), 1 < p < ∞. THEOREM: Let G be an infinite LCA group. Then there exists a strongly non regular Fourier multiplier on Lp(G), 1 < p < ∞. PARTS OF THE PROOF: The General Structure Theorem for LCA groups says that G is isomorphic with Rn × G0 with n ≥ 0 and G0 is an LCA group containing a compact subgroup K such that G0/K is discrete. Distinguish 3 cases: 1.) if n ≥ 1, then use the Hilbert transform on R and the structure proposition above. 2.) If n = 0, then G ∼ = G0. If K is infinite, then use the above proposition for infinite compact groups. 3.) If K is finite, then G0 itself must be discrete, so use the above proposition for infinite discrete groups.

Property (P)

Let T : M → M be a w∗ continuous operator. T is said to satisfy property (P) if there exist v1, v2 : M → M such that v1 T T ◦ v2

• : M2(M) → M2(M)

is completely positive, completely contractive and self-adjoint. REMARK: If T satisfies property (P), then T is contractively decomposable and self-adjoint, but the converse fails. THEOREM [A.-K.]: Let G = Fn be the free group. Let Tm : VN(G) → VN(G) be a Fourier multiplier satisfying (P). Let 1 < p < ∞. Then Tm satisfies the noncommutative Matsaev’s

• inequality. More precisely, for any complex polynomial P, we have

P(Tm)cb,Lp→Lp ≤ P(S)cb,ℓp

Z→ℓp Z,

where S is the unitary right shift on ℓp

Z.

Property (P)

What are the Fourier multipliers satisfying property (P) ? CONJECTURE: If T : VN(Fn) → VN(Fn) is a block-radial Fourier multiplier, i.e. if f = λgk1

i1

. . . λg

kN iN

with i1 = i2 = . . . = iN, then Tf = φ(N)f , then T satisfies (P) if and only if T is completely contractive and φ is real-valued (i.e. T self-adjoint).

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