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Quantum information theory via Fourier

multipliers on quantum groups

Cedric Arhancet

Abstract

In this paper, we compute the exact values of the minimum output entropy and thecompletely bounded minimal entropy of quantum channels acting on matrix algebras Mn

belonging to large and infinite classes of quantum channels. We also give a upper boundof the classical capacity which is an equality in some cases. For that, we use the theoryof quantum groups and our results rely on a new and precise description of boundedFourier multipliers from L1(G) into Lp(G) for 1 < p 6 ∞ where G is a co-amenable locallycompact quantum group of Kac type and on the automatic completely boundedness ofthese multipliers that this description entails. We also connect egrodic actions of (quantum)groups to the topic of computation of the minimum output entropy and the completelybounded minimal entropy of channels.

Contents

1 Introduction 2

2 Preliminaries 5

3 Convolutions by elements of Lp(G) 11

3.1 Convolutions by elements of Lp(G) and c.b. maps from L1(G) into Lp(G) . . . . 113.2 Bounded versus completely bounded Fourier multipliers from L1(G) into Lp(G) . 143.3 Another approach on c. b. Fourier multipliers from L1(G) into Lp(G) . . . . . . 193.4 Bounded versus completely bounded operators from L1(M) into Lp(M) . . . . . 223.5 ℓp(Spd)-summing Fourier multipliers on L∞(G) . . . . . . . . . . . . . . . . . . . . 26

4 Entropies and capacities 32

4.1 Definitions and transference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Entropies and classical capacities of Fourier multipliers . . . . . . . . . . . . . . . 35

5 Examples 38

5.1 Kac-Paljutkin quantum group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Dihedral groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 The principle of transference in noncommutative analysis 43

6.1 Actions of locally compact quantum groups . . . . . . . . . . . . . . . . . . . . . 436.2 Transference I : from Fourier multipliers to Herz-Schur multipliers . . . . . . . . 466.3 Transference II: from Fourier multipliers to channels via representations . . . . . 50

7 Entanglement breaking and PPT Fourier multipliers 50

7.1 Multiplicative domains of Markov maps . . . . . . . . . . . . . . . . . . . . . . . 507.2 PPT and entangling breaking maps . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Mathematics subject classification: 20G42, 43A22, 46L07, 46L52, 81P45.Key words and phrases: noncommutative Lp-spaces, quantum entropy, quantum capacities, compact quantumgroups, Fourier multipliers, quantum information theory.

1

8 Appendices 54

8.1 Appendix: output minimal entropy as derivative of operator norms . . . . . . . . 548.2 Appendix: multiplicativity of some completely bounded operator norms . . . . . 56

Bibliography 60

1 Introduction

One of the most fundamental questions in quantum information explicitly stated in [Sho1]concerns with the amount of information that can be transmitted reliably through a quantumchannel. For that, many capacities was introduced for describing the capability of the channelfor delivering information from the sender to the receiver. Determine these quantities is centralto characterizing the limits of the quantum channel’s ability to transmit information reliably.

In the Schodinger picture, a quantum channel is a trace preserving completely positive mapT : S1

n → S1n acting on the Schatten trace class space S1

n. The minimum output entropy of aquantum channel is defined by

(1.1) Hmin(T )def= min

x>0,Trx=1H(T (x))

where H(ρ)def= − Tr(ρ log ρ) denotes the von Neumann entropy. Recall that H is the quantum

analogue of the Shannon entropy and can be seen as a uncertainty measure of ρ. See for example[Pet2] for a short survey of this notion. Note that H(ρ) is always positive, takes its maximumvalues when ρ is maximally mixed and its minimum value zero if ρ is pure. So the minimumoutput entropy can be seen as a measure of the deterioration of the purity of states. Theminimum output entropy of a specific channel is difficult to calculate in general.

In this work, we investigate more generally quantum channels acting on a finite-dimensionalnoncommutative L1-space L1(M) associated with a finite-dimensional von Neumann algebra1

M equipped with a faithful trace τ . The first aim of this paper is to compute the quantity

(1.2) Hmin(T )def= min

x>0,τ(x)=1H(T (x))

for any T : L1(M) → L1(M) belonging to a large class of explicit quantum channels. Here

H(ρ)def= −τ(ρ log ρ) denotes the Segal entropy introduced in [Seg1]. Note that if the trace τ

is normalized and if ρ is a positive element of L1(M) with τ(ρ) = 1 then H(ρ) 6 0. We refer[LuP1], [LPS1], [NaU1], [OcS1] and [Rus1] for more information on the Segal entropy.

The considered channels are the (trace preserving completely positive) Fourier multiplierson finite quantum groups, see e.g. [Daw1] and [Daw3] for more information. Such a chan-nel T : L1(G) → L1(G) acts on a finite-dimensional noncommutative L1-space L1(G) and is anoncommutative variant of a classical Fourier multiplier Mϕ : L1(T) → L1(T),

∑k ake2πik· 7→

∑k ϕkake2πik· where ϕ = (ϕk) is a suitable sequence of ℓ∞

Z. Here T

def= {z ∈ C : |z| = 1} is

the one-dimensional torus and we refer to [Gra1] for the classical theory of multipliers on T.Loosely speaking, a finite quantum group is a finite-dimensional C∗-algebra A equipped witha suitable “coproduct” map ∆: A → A ⊗ A that is a noncommutative variant of the commu-tative C∗-algebra C(G) of continuous functions on a compact group G equipped with the map∆: C(G) → C(G) ⊗ C(G) where ∆(f)(s, t) = f(st) for f ∈ C(G) and s, t ∈ G. It is possibleto do harmonic analysis and representation theory in this setting. See [FSS1], [KuV2], [MaD1]and [Maj1] for short presentations.

1. Such a algebra M is isomorphic to a direct sum of matrix algebras Mn1 ⊕ · · · ⊕ MnK.

2

Note that we can see our channels as acting on matrix algebras by using a suitable ∗-homomorphism J and a conditional expectation E, see Section 5.

Our approach relies on an observation of the authors of [ACN1]. In the case M = Mn, theyobserved that we can express the minimum output entropy as a derivative of suitable operatorsnorms:

(1.3) Hmin(T ) = −d

dp‖T ‖pL1(M)→Lp(M) |p=1.

See Section 8.1 for a proof of this formula for the slightly more general case of finite-dimensionalvon Neumann algebras. So it suffices to compute the operator norm ‖·‖L1(M)→Lp(M) in orderto compute the minimum output entropy. Unfortunately, in general, the calculation of theseoperator norms is not much easier than the direct computation of the minimum output entropy.However, the first purpose of this paper is to show that the computation of ‖T ‖L1(G)→Lp(G) ispossible in the case of an arbitrary Fourier multiplier T on a finite quantum group G and moregenerally on a co-amenable compact quantum group of Kac type.

Moreover, we prove that for these channels the minimum output entropy is equal to thecompletely bounded minimal entropy (reverse coherent information). This notion was intro-duced for M = Mn in [DJKRB] and rediscovered in [GPLS], see also [YHW1]. We can use thefollowing definition which will be justified in Theorem 8.12

(1.4) Hcb,min(T )def= −

d

dp‖T ‖pcb,L1(M)→Lp(M) |p=1

where the subscript cb means completely bounded and is natural in operator space theory, seethe books [BLM], [ER], [Pau], [Pis1] and [Pis2]. We refer again to Theorem 8.12 for a concreteexpression of Hcb,min(T ). Indeed, we will show that ‖T ‖cb,L1(G)→Lp(G) = ‖T ‖L1(G)→Lp(G) forany bounded Fourier multiplier T on a co-amenable compact quantum group of Kac type G(hence in particular on finite quantum groups) and any 1 6 p 6 ∞ (see Theorem 3.9) andwe will immediately deduce the equality Hmin(T ) = Hcb,min(T ). Note that if M is a non-abelian von Neumann algebra, we will prove in Section 3.4 that the completely bounded norm‖·‖cb,L1(M)→Lp(M) and the classical operator norm ‖·‖L1(M)→Lp(M) are not equal. So the aboveequality is specific to the Fourier multipliers on co-amenable compact quantum groups of Kactype. Finally, we refer to [GuW1] for another application of these completely bounded norms.

Note that the completely bounded minimal entropy Hcb,min is additive (see Theorem 8.11)in sharp contrast with the minimum output entropy [Has1].

Fundamental characteristics of a quantum channel are the different capacities. We refer tothe survey [GIN1] for more information on the large diversity of quantum channel capacities.We refer to the survey [GIN1] for more information on the large diversity of quantum channelcapacities. Despite of the significant progress in recent years, the computation of most capacitiesremains largely open. The Holevo capacity of a quantum channel T : L1(M) → L1(M) can bedefined by the formula

(1.5) χ(T )def= sup

{H

( N∑

i=1

λiT (ρi)

)−

N∑

i=1

λiH(T (ρi))

}

where the supremum is taken over all N ∈ N, all probability distributions (λ1, . . . , λN ) and allstates ρ1, . . . , ρN of L1(M), see e.g. [GIN1, page 21] and [Key1, page 82] for the case M = Mn.It is known that the classical capacity C(T ) can be expressed by the asymptotic formula

(1.6) C(T ) = limn→+∞

χ(T⊗n)

n.

3

This quantity is the rate at which one can reliably send classical information through T , seeDefinition 4.1. Using our computation of Hmin(T ), we are able to determine a sharp upperbound of C(T ) for any Fourier multiplier T acting on a finite quantum group G (see Theorem4.14). It is the second aim of this paper. Recall that the classical capacity is always biggerthan the quantum capacity Q(T ) which the the rate at which one can reliably send quantuminformation through T , see Definition 4.1.

We refer to [GJL1], [GJL2], [JuP1] and [JuP2] for papers using operator space theory forestimating quantum capacities and to the books [GMS1], [Hol1], [NiC1], [Pet1], [Wat1] and[Wil1] for more information on quantum information theory.

To conclude, we also hope that this connection between quantum information theory andthe well-established theory of Fourier multipliers on operator algebras will be the starting pointof other progress on the problems of quantum information theory but also in noncommutativeharmonic analysis. We leave open an important number of implicit questions. We equallyrefer to [BCLY], [CrN1] and [LeY1] for papers on quantum information theory in link withnoncommutative harmonic analysis. See also [ACN2] and [NSSS1] for very recent paper onFourier multipliers on quantum groups.

The paper is organized as follows. Section 2 gives background on noncommutative Lp-spaces, compact quantum groups and operator space theory. These notions are fundamental forus. In Section 3.1, we prove that the convolution induces a completely bounded map on somenoncommutative Lp-spaces associated to some suitable compact quantum groups. In Section3.2, we describe bounded Fourier multipliers from L1(G) into Lp(G) where G is a co-amenablecompact quantum group of Kac type. We deduce from Section 3.1 that these multipliers arenecessarily completely bounded with the same completely bounded norm. We also describe byduality the (completely) bounded Fourier multiplier Lp(G) → L∞(G) which will be useful forobtaining the result on the classical capacities. In Section 3.3, we describe some completelybounded Fourier multipliers on compact quantum groups of Kac type beyond the coamenablecase.

In Section 3.4, we compare the spaces of bounded maps and of completely bounded mapsfrom L1(M) into L∞(M). We show that in the case of a nonabelian von Neumann algebra M,these norms are different. We equally examine observe the completely bounded Schur multipliesfrom S1 into the space B(ℓ2) of bounded operators acting on the Hilbert space ℓ2.

In Section 3.5, we examine the ℓp(Spd)-summing (left or right) Fourier multipliers acting fromL∞(G) into L∞(G) where G is a co-amenable compact quantum group of Kac type. In Section4.1, we investigate the relations between the capacities of a channel T : L1(M) → L1(M) andits extension JTE : L1(N ) → L1(N ) where J : M → N is a trace preserving unital normal∗-injective homomorphism and where E is the associated conditional expectation. In Section4.2, we compute the minimal output entropy and we obtain the equality with the completelybounded minimal entropy. Finally, we obtain our sharp upper bound on the classical capacity.

In Section 5.2, 5.1, we illustrate our results with the Kac-Paljutkin quantum group anddihedral groups.

In Section 6.1 and the next sections, we show how our results on quantum channels onoperator algebras allows us to recover by transference some results of [GJL1] and [GJL2] forquantum channels T : S1

n → S1n acting on the Schatten trace class space S1

n.In Section 7.1, we give a description of multiplicative domains of Markov maps (=quantum

channels) and describe multiplicative domains of Fourier multipliers. In section 7.2, we char-acterize PPT conditional expectations and entangling breaking PPT conditional expectations(Corollary 7.7). We also give a necessary condition on PPT Fourier multipliers.

In Section 8.1, we give a proof of the formula (1.3) which describes the minimum outputentropy Hmin(T ) as a derivative of operator norms. In Section 8.2, we give a proof of the

4

multiplicativity of the completely bounded norms ‖·‖cb,Lq→Lp if 1 6 q 6 p 6 ∞. We deducethe additivity of Hcb,min(T ).

2 Preliminaries

Noncommutative Lp-spaces Here we recall some facts on noncommutative Lp-spaces as-sociated with semifinite von Neumann algebras. We refer to [Tak1] [Str] for the theory of vonNeumann algebras and to [PiX] for more information on noncommutative Lp-spaces. Let Mbe a semifinite von Neumann algebra equipped with a normal semifinite faithful trace τ . Wedenote by M+ the positive part of M. Let S+ be the set of all x ∈ M+ whose support projec-tion have a finite trace. Then any x ∈ S+ has a finite trace. Let S ⊂ M be the complex linearspan of S+, then S is a weak*-dense ∗-subalgebra of M.

Let 1 6 p < ∞. For any x ∈ S, the operator |x|p belongs to S+ and we set

(2.1) ‖x‖Lp(M)

def=(τ(|x|p)

) 1p .

Here |x|def= (x∗x)

12 denotes the modulus of x. It turns out that ‖·‖Lp(M) is a norm on S. By

definition, the noncommutative Lp-space Lp(M) associated with (M, τ) is the completion of

(S, ‖·‖Lp(M)). For convenience, we also set L∞(M)def= M equipped with its operator norm.

Note that by definition, Lp(M) ∩ M is dense in Lp(M) for any 1 6 p < ∞.Furthermore, τ uniquely extends to a bounded linear functional on L1(M), still denoted by

τ . Indeed, we have

(2.2) |τ(x)| 6 ‖x‖L1(M) , x ∈ L1(M).

We recall the noncommutative Holder’s inequality. If 1 6 p, q, r 6 ∞ satisfy 1r

= 1p

+ 1q

then

(2.3) ‖xy‖Lr(M) 6 ‖x‖Lp(M) ‖y‖Lq(M) , x ∈ Lp(M), y ∈ Lq(M).

Conversely for any z ∈ Lr(M), by using the same ideas than the proof of [Ray2, Theorem 4.20]there exist x ∈ Lp(M) and y ∈ Lq(M) such that

(2.4) z = xy with ‖z‖Lr(M) = ‖x‖Lp(M) ‖y‖Lq(M) .

For any 1 6 p < ∞, let p∗ def= p

p−1 be the conjugate number of p. Applying (2.3) with q = p∗

and r = 1 together with (2.2), we obtain a map Lp∗

(M) → (Lp(M))∗, y 7→ τ(xy) which inducesan isometric isomorphism

(2.5) (Lp(M))∗ = Lp∗

(M), 1 6 p < ∞,1

p+

1

p∗= 1.

In particular, we may identify L1(M) with the unique predual M∗ of M. Finally, for anypositive element in Lp(M), recall that there exists a positive element y ∈ Lp

∗

(M) such that

(2.6) ‖y‖Lp∗ (M) = 1 and ‖x‖Lp(M) = τ(xy).

5

Locally compact quantum groups Let us recall some well-known definitions and propertiesconcerning compact quantum groups. We refer to [Cas1], [EnS1], [FSS1], [Kus1], [MaD1],[NeT1], [VDa1] and references therein for more details.

A locally compact quantum group is a pair G = (L∞(G),∆) with the following properties:

1. L∞(G) is a von Neumann algebra,

2. ∆: L∞(G) → L∞(G)⊗L∞(G) is a co-multiplication, that is an injective faithful normalunital ∗-homomorphism which is co-associative:

(2.7) (∆ ⊗ Id)∆ = (Id ⊗ ∆)∆,

3. there exist normal semifinite faithful weights ϕ, ψ on L∞(G), called called the left andright Haar weights satisfying

ϕ((ω ⊗ Id)∆(x)) = ω(1)ϕ(x) for all ω ∈ L∞(G)+∗ , x ∈ L∞(G)+ such that ϕ(x) < ∞,

ψ((Id ⊗ ω)∆(x)) = ω(1)ψ(x) for all ω ∈ L∞(G)+∗ , x ∈ L∞(G)+ such that ψ(x) < ∞.

Let G be a locally compact quantum group. By [Kus1, Theorem 5.2], the left and rightHaar weights, whose existence is assumed, are unique up to scaling. If the left and right Haarweights ϕ and ψ of G coincide then we say that G is unimodular.

Let Nϕdef= {x ∈ L∞(G) : ϕ(x∗x) < ∞}. By [Kus1, Theorem 5.4], there exists a unique

σ-strongly* closed linear operator S : domS ⊂ L∞(G) → L∞(G) such that the linear space

(2.8)⟨(Id ⊗ ϕ)(∆(z)(1 ⊗ y)) : y, z ∈ Nϕ

⟩

is a core for S with respect to the σ-strong* topology and satisfying

(2.9) S((Id ⊗ ϕ)(∆(z)(1 ⊗ y))

)= (Id ⊗ ϕ)

((1 ⊗ z)∆(y)

), y, z ∈ Nϕ.

We say that S is the antipode of L∞(G). By [Kus1, Proposition 5.5] there exists a unique∗-antiautomorphism R : L∞(G) → L∞(G) and a unique weak* continuous group τ = (τt)t∈R of∗-automorphisms such that2

(2.10) S = Rτ− i2, R2 = Id and τtR = Rτt, for any t ∈ R.

We say thatR is the unitary antipode and that τ is the scaling group of G. If Σ: L∞(G)⊗L∞(G) →L∞(G)⊗L∞(G), x⊗ y 7→ y ⊗ x is the flip, by [Kus1, Proposition 5.6] we have

(2.11) ∆ ◦R = Σ ◦ (R⊗R) ◦ ∆.

Since ϕ ◦R is a right invariant weight, we can suppose that

(2.12) ψ = ϕ ◦R.

If (σt)t∈R is the modular automorphism group associated to the left Haar weight ϕ, the locallycompact quantum group G is said to be of Kac type (or a Kac algebra) if G has trivial scalinggroup and Rσt = σ−tR for any t ∈ R. In this case, we have

(2.13) S = R.

2. Note that the equation S = Rτ− i2

has to be understood in the following sense: when x ∈ L∞(G) is analytic

with respect to τ , then x belongs to the domain of S and S(x) = Rτ− i2

(x).

6

We refer to [VaV1, page 7] for other characterizations of Kac algebras.Following [Run1, Proposition 3.1], we say that a locally compact quantum group G is com-

pact if the left Haar weight ϕ of G is finite. Such group is always unimodular and the cor-responding Haar weight can always be chosen to be a state h : L∞(G) → C (called the Haarstate). In this case, by (2.12) R preserves the finite trace h, i.e. h ◦R = h. Moreover, we haveby essentially [KuV1, page 14]

(2.14) (h⊗ Id) ◦ ∆(x) = h(x)1 = (Id ⊗ h) ◦ ∆(x), x ∈ L∞(G).

For a compact quantum group G being of Kac type is equivalent to the traciality of the Haarstate h, often denoted by τ . This is equivalent to the fact that its dual G defined in [Kus1,Theorem 5.4] is unimodular.

We say that G is finite if L∞(G) is finite-dimensional. Such a compact quantum group is ofKac type by [VDa2].

Representations Suppose that G is a compact quantum group. A unitary matrix u =[uij ]16i,j6n element of Mn(L∞(G)) with entries uij ∈ L∞(G) is called a n-dimensional unitaryrepresentation of G if for any i, j = 1, . . . , n we have

(2.15) ∆(uij) =

n∑

k=1

uik ⊗ ukj .

We say that u is irreductible if there is no non-trivial projection p ∈ Mn such that (p ⊗1)u = u(p ⊗ 1). We denote by Irr(G) the set of unitary equivalence classes of irreduciblefinite-dimensional unitary representations of G. For each π ∈ Irr(G), we fix a representativeu(π) ∈ L∞(G) ⊗ Mnπ

of the class π.We let

(2.16) Pol(G)def= span

{u

(π)ij : u(π) = [u

(π)ij ]nπ

i,j=1, π ∈ Irr(G)}.

This is a weak* dense subalgebra of L∞(G).

Antipodes on compact quantum groups of Kac type If G is a compact quantum groupof Kac type, it is known that the antipode of G is an anti-∗-automorphism R : L∞(G) → L∞(G)determined by

R(u

(π)ij

)=(u

(π)ji

)∗, π ∈ Irr(G), 1 6 i, j 6 nπ.

Fourier transforms Suppose that G is a compact quantum group of Kac type. For a linearfunctional ϕ on Pol(G), we define the Fourier transform ϕ = (ϕ(π))π∈Irr(G), which is an element

of ℓ∞(G)def= ⊕πMnπ

, by

ϕ(π)def= (ϕ⊗ Id)

((u(π))∗

), π ∈ Irr(G).

In particular, any x ∈ L∞(G) induces a linear functional h(·x) : Pol(G) → C, y 7→ h(yx) onPol(G) and the Fourier transform x = (x(π))π∈Irr(G) of x is defined by

x(π)def= (h(·x) ⊗ Id)

((u(π))∗

), π ∈ Irr(G).

7

Let ϕ1, ϕ2 be linear functionals on Pol(G). Consider their convolution product ϕ1 ∗ ϕ2def=

(ϕ1 ⊗ ϕ2) ◦ ∆. We recall that

(2.17) ϕ1 ∗ ϕ2(π) = ϕ2(π)ϕ1(π).

The following result is elementary and well-known.

Proposition 2.1 The Fourier transform F : L∞(G)∗ → ℓ∞(G), f 7→ f is a contraction, andmoreover F sends L1(G) injectively into ℓ∞(G).

Example 2.2 Let G be a locally compact group and define

(2.18) ∆(f)(s, t)def= f(st), f ∈ C(G), s, t ∈ G.

Then ∆ admits a weak* continuous extension ∆: L∞(G) → L∞(G)⊗L∞(G). If µG is a leftHaar measure on G, then a left Haar weight ϕ on the abelian von Neumann algebra L∞(G)is given by ϕ(f) =

∫Gf dµG and similarly we obtain a right Haar weight ψ with a right Haar

measure. In this situation, L∞(G) equipped with ∆, ϕ and ψ is a locally compact quantumgroup which is compact if and only if the group G is compact. The antipode S is given by(Sf)(s) = f(s−1) where s ∈ G. If G is compact, for any f ∈ L∞(G), we can show that

f(π) =

∫

G

π(s)∗f(s) dµG(s), π ∈ Irr(G).

Example 2.3 Let G be a locally compact group with its neutral element e and VN(G) itsvon Neumann algebra generated by the left translation operators {λs : s ∈ G} on L2(G). The“dual” G = G of G is a locally compact quantum group defined by VN(G) equipped with thecoproduct ∆: VN(G) → VN(G)⊗VN(G) defined by

(2.19) ∆(λs) = λs ⊗ λs.

The left and right Haar weights of G are both equal to the Plancherel weight of G [Tak2, SectionVII.3].

In the case where G is discrete, this quantum group is compact and the Haar state is definedas the unique trace τ on VN(G) such that τ(1) = 1 and τ(λs) = 0 for s ∈ G \ {e}. For anyf ∈ VN(G), we have

f(s) = τ(fλ(s)∗), s ∈ G.

The antipode R : VN(G) → VN(G) is defined by S(λs) = λs−1 .

Operator spaces We refer to the books [BLM], [ER], [Pau] and [Pis2] for background onoperators spaces. We denote by ⊗min and ⊗ the injective and the projective tensor product ofoperator spaces. If E and F are operator spaces, we let CB(E,F ) for the space of all completelybounded maps endowed with the norm

‖T ‖cb,E→F = supn>1

∥∥IdMn⊗ T

∥∥Mn(E)→Mn(F )

.

If E and F are operator spaces, [BLM, (1.32)] we have a complete isometry

(2.20) E ⊗min F ⊂ CB(F ∗, E)

8

and each element of x ⊗ y of E ⊗min F identifies to a weak*-weak continuous map (see [DeF,page 48]. Moreover, recall the complete isometry of [BLM, page 27]

(2.21) E∗ ⊗min F ⊂ CB(E,F ).

Let M and N be von Neumann algebras. Recall that by [Pis2, Theorem 2.5.2] [BLM, page 40]the map

(2.22)Ψ: M⊗N −→ CB(M∗,N )

z 7−→ f 7→ (f ⊗ Id)(z)

is a completely isometric isomorphism. Let X be a Banach space and E be an operator space.For any bounded linear map T : X → E, we have by [BLM, (1.12)]

(2.23) ‖T ‖cb,max(X)→E = ‖T ‖X→E .

For any bounded linear map T : E → X , we have by [BLM, (1.10)]

(2.24) ‖T ‖cb,E→min(X) = ‖T ‖E→X .

Recall that by [Pau, Exercise 3.10 (ii)] if T : → F is a bounded map between operator spaceswe have

(2.25) ‖IdMd⊗ T ‖Md(E)→Md(F ) 6 d ‖T ‖E→F .

Vector-valued noncommutative Lp-spaces The theory of vector-valued noncommutativeLp-spaces was initiated by Pisier [Pis1] for the case where the underlying von Neumann algebraM is hyperfinite and equipped with a normal semifinite faithful trace. Under these assumptions,according to [Pis1, pages 37-38], for any operator space E, the operator spaces M ⊗min E andMop

∗ ⊗E can be injected into a common topological vector space. This compatibility in thesense of interpolation theory, explained in [Pis1, page 37] and [Pis2, page 139] relies heavily onthe fact that the von Neumann algebra M is hyperfinite. Suppose 1 6 p 6 ∞. Then we candefine by complex interpolation the operator space

(2.26) Lp(M, E)def=(M ⊗min E,M

op∗ ⊗E

)1p

.

When E = C, we get the noncommutative Lp-space Lp(M) defined by (2.1).Let M be a hyperfinite von Neumann algebra. Let E and F be operator spaces. If a map

T : E → F is completely bounded then by [Pis1, (3.1) page 39] it induces a completely boundedmap IdLp(M) ⊗ T : Lp(M, E) → Lp(M, F ) and we have

(2.27)∥∥IdLp(M) ⊗ T

∥∥cb,Lp(M,E)→Lp(M,F )

6 ‖T ‖cb,E→F .

If M is hyperfinite, we will use sometimes the following notation of [Pis1, Definition 3.1]3:

(2.28) L∞(M, E)def= M ⊗min E.

Let M and N be hyperfinite von Neumann algebras equipped with normal semifinite faithfultraces. Suppose 1 6 q 6 p 6 ∞. We have a completely bounded map F : Lq(M,Lp(N )) →Lp(N ,Lq(M)), x⊗ y 7→ y ⊗ x (the flip) with

(2.29) ‖F‖cb,Lq(M,Lp(N ))→Lp(N ,Lq(M)) 6 1.

3. It would be more correct to use the notation L∞0 (M, E)

9

If M and N are von Neumann algebras equipped with normal semifinite faithful traces andif 1 6 q 6 p 6 ∞ and 1

q= 1

p+ 1

r, we introduce the norm on Lp(M) ⊗ Lq(N ) [JP1, page 71]

(2.30) ‖x‖Lp(M,Lq(N )) = sup‖a‖L2r (M),‖b‖L2r (M)61

∥∥(a⊗ 1)x(b⊗ 1)∥∥

Lq(M⊗N ).

If M is hyperfinite, this norm coincide with the norm of the space Lp(M,Lq(N )) defined by(2.26) with E = Lq(N ). If x > 0, we can take the infimum with a = b. In particular, if1 6 p 6 ∞, we have

(2.31) ‖x‖L∞(M,Lp(N )) = sup‖a‖L2p(M),‖b‖L2p(M)61

∥∥(a⊗ 1)x(b ⊗ 1)∥∥

Lp(M⊗N ).

By [GJL3, lemma 3.11] and [JP1, Lemma 4.9], if E = Id ⊗ τ : M⊗N → M is the canonicaltrace preserving normal conditional expectation, we have

(2.32) ‖x‖L∞(M,L1(N )) = infx=x1x2

∥∥E(x1x∗1)∥∥ 1

2

L∞(M)

∥∥E(x∗2x2)

∥∥ 12

L∞(M).

Multipliers and bimodules Let A be a Banach algebra and let X be a Banach spaceequipped with a structure of A-bimodule. We say that a pair of maps (L,R) from A into X isa multiplier if

(2.33) aL(b) = R(a)b, a, b ∈ A.

We say that (L,R) is a bounded multiplier if L and R are bounded. In this case, we let

(2.34) ‖(L,R)‖def= max

{‖L‖A→X , ‖R‖A→X

}.

If X is A-right module, a left multiplier is a map L : A → X such that

(2.35) L(ab) = L(a)b, a, b ∈ A.

Similarly, if X is A-left module, a right multiplier is a map R : A → X such that

(2.36) R(ab) = aR(b), a, b ∈ A.

We refer to [Daw1] for more information.

Matrix normed modules Let A be a Banach algebra equipped with an operator spacestructure. We need to consider the left A-modules X that correspond to completely contractivehomomorphisms A → CB(X) for an operator space X . Following [BLM, (3.1.4)], we call thesematrix normed left A-modules. It is easy to see that these are exactly the left A-modules Xsatisfying for all m,n ∈ N, any [aij ] ∈ Mn(A) and any [xkl] ∈ Mm(X)

(2.37)∥∥[aijxkl]

∥∥Mnm(X)

6∥∥[aij ]

∥∥Mn(A)

∥∥[xkl]∥∥

Mm(X).

A reformulation of (2.37) is that the module action on X extends to a complete contractionA⊗X → X where ⊗ denotes the operator space projective tensor product. Similar results holdfor matrix normed right modules, e.g. the right analogue of (2.37) is

(2.38)∥∥[xklaij ]

∥∥Mnm(X)

6∥∥[aij ]

∥∥Mn(A)

∥∥[xkl]∥∥

Mm(X).

If X is a matrix normed left A-module, then by [BLM, 3.1.5 (2)] the dual X∗ is a matrix normedright A-module for the dual action defined by

(2.39) 〈ϕa, x〉X∗,Xdef= 〈ϕ, ax〉X∗,X , ϕ ∈ X∗, a ∈ A, x ∈ X.

There exists a similar result for matrix normed right A-modules.

10

Entropy Let g : R+ → R be the function defined by g(0)def= 0 and g(t)

def= t ln(t) if t > 0.

Let M be a von Neumann algebra with a normal faithful finite trace τ . If ρ is positive element

of L1(M), by a slight abuse, we use the notation ρ log ρdef= g(ρ). In this case, if in addition

‖ρ‖L1(M) = 1 we can define the Segal entropy H(ρ)def= −τ(ρ log ρ) introduced in [Seg1]. If we

need to specify the trace τ , we will use the notation H(ρ, τ). Recall that by [OcS1, page 78], ifρ1, ρ2 are positive elements of L1(M) with norm 1 (i.e states), we have

(2.40) H(ρ1 ⊗ ρ2) = H(ρ1) + H(ρ2).

Recall that for any t > 0 and any state ρ ∈ L1(M, τ) and any state ρ1 ∈ L1(M, tτ), we have

(2.41) H(ρ, τ) = H

(1

tρ, tτ

)− log t and H(tρ1, τ) = H(ρ1, tτ) − log t.

Elementary results Let f : [0,+∞) → R+ be a function with limq→∞ f(q) = 1. By [JuP1,Lemma 2.6], exists if and only if limq→∞ q ln f(q) exists. In this case, we have4

(2.42)d

dpf(p∗)|p=1 = lim

q→∞q ln f(q).

Consider a net (fα) of continuous functions fα : K → R on a compact topological space K.Suppose that (fα) converges uniformly. Then it is easy to check that

(2.43) limα

minx∈K

fα(x) = minx∈K

limαfα(x).

3 Convolutions by elements of Lp(G)

3.1 Convolutions by elements of Lp(G) and completely bounded maps

from L1(G) into Lp(G)

Let G be a locally compact quantum group. Recall that the associated coproduct is a weak*continuous completely contractive map ∆: L∞(G) → L∞(G)⊗L∞(G) and that by [ER, Theo-rem 7.2.4] we have (L∞(G)⊗L∞(G))∗ = L∞(G)∗⊗L∞(G)∗ where ⊗ denotes the operator spaceprojective tensor product. So the preadjoint map ∆∗ : L∞(G)∗⊗L∞(G)∗ → L∞(G)∗ is com-pletely contractive. This gives us a completely contractive Banach algebra structure on thepredual L∞(G)∗. We will use the following classical notation

(3.1) ω1 ∗ ω2def= ∆∗(ω1 ⊗ ω2) = (ω1 ⊗ ω2) ◦ ∆, ω1, ω2 ∈ L∞(G)∗.

Suppose that G is of Kac type. Suppose 1 6 p 6 ∞. For any f ∈ L1(G) and any g ∈ Lp(G),the authors of [LWW1, Corollary 3.10]5 obtain a Young’s inequality

(3.2) ‖f ∗ g‖Lp(G) 6 ‖f‖L1(G) ‖g‖Lp(G) .

where ∗ is a suitable extension of the above convolution (3.1).In this section, we prove a completely bounded version of this inequality.

4. Recall that p∗ = pp−1

.

5. The result remains true for a locally compact quantum group G whose scaling automorphism group is trivial.

11

Note that (3.1), induces a left action and a right action of L∞(G)∗ on L∞(G)∗. We definethe right dual action ⋆ of L∞(G)∗ on L∞(G) by

(3.3) 〈x ⋆ ω1, ω2〉L∞(G),L∞(G)∗

(2.39)= 〈x, ω1 ∗ ω2〉L∞(G),L∞(G)∗

, ω1, ω2 ∈ L∞(G)∗, x ∈ L∞(G)

and the left dual action ⋆ of L∞(G)∗ on L∞(G) by

(3.4) 〈ω1 ⋆ x, ω2〉L∞(G),L∞(G)∗

(2.39)= 〈x, ω2 ∗ ω1〉L∞(G),L∞(G)∗

, ω1, ω2 ∈ L∞(G)∗, x ∈ L∞(G).

Lemma 3.1 For any x ∈ L∞(G) and any ω ∈ L∞(G)∗, we have

(3.5) x ⋆ ω = (ω ⊗ Id) ◦ ∆(x) and ω ⋆ x = (Id ⊗ ω) ◦ ∆(x)

Proof : For any x ∈ L∞(G) and any ω, ω0 ∈ L∞(G)∗, we have

〈x ⋆ ω, ω0〉L∞(G),L∞(G)∗

(3.3)= 〈x, ω ∗ ω0〉L∞(G),L∞(G)∗

(3.1)= (ω ⊗ ω0) ◦ ∆(x) = ω0

((ω ⊗ Id) ◦ ∆(x)

)

=⟨(ω ⊗ Id) ◦ ∆(x), ω0

⟩L∞(G),L∞(G)∗

and similarly

〈ω ⋆ x, ω0〉L∞(G),L∞(G)∗

(3.4)= 〈x, ω0 ∗ ω〉L∞(G),L∞(G)∗

(3.1)= (ω0 ⊗ ω) ◦ ∆(x) = ω0

((Id ⊗ ω) ◦ ∆(x)

)

=⟨(Id ⊗ ω) ◦ ∆(x), ω0

⟩L∞(G),L∞(G)∗

.

We deduce (3.5) by duality.

Recall that the Banach space L1(G) is equipped with the operator space structure inducedby the isometry i : L1(G) → L∞(G)op

∗ , f 7→ τ(f ·) and the opposed operator space structureL∞(G)op

∗ of L∞(G)∗, i.e. this map is completely isometric. See [Pis2, page 139] for a discussionof the opposite structure in the context of noncommutative Lp-spaces.

For any y ∈ L1(G) and any x ∈ L∞(G), we define the elements y ∗1,∞ x and x ∗∞,1 y ofL∞(G) by

(3.6) y ∗1,∞ xdef= x ⋆ τ(yR(·)) and x ∗∞,1 y

def= τ(yR(·)) ⋆ x.

By (3.5), we have

(3.7) y ∗1,∞ x =(τ(yR(·)) ⊗ Id

)◦ ∆(x) and x ∗∞,1 y =

(Id ⊗ τ(yR(·))

)◦ ∆(x).

In the following proposition, the intersection L∞(G) ∩ L1(G) is obtained with respect to theembeddings of L∞(G) and L1(G) in the space L0(G) of measurable operators.

Proposition 3.2 Let G be a locally compact quantum group of Kac type. We have the followingcommutative diagram.

L∞(G)∗⊗L∞(G)∗∗ // L∞(G)∗

L1(G) ⊗ (L∞(G) ∩ L1(G))∗1,∞

//?�

i⊗i

OO

L∞(G) ∩ L1(G)?�

i

OO

12

Proof : For any z ∈ L∞(G), any x ∈ L∞(G) ∩ L1(G) and any y ∈ L1(G), we have on the onehand

⟨i(y ∗1,∞ x), z

⟩L∞(G)∗,L∞(G)

(3.7)=⟨i((τ(yR(·)) ⊗

)◦ ∆(x)

), z⟩

L∞(G)∗,L∞(G)

= τ([(

τ(yR(·)) ⊗ Id)

◦ ∆(x))]z)

=(τ(yR(·)) ⊗ τ(z ·)

)◦ ∆(x).

On the other hand, we have

⟨i(y) ⊗ i(x), z

⟩L∞(G)∗,L∞(G)

=⟨τ(y ·) ∗ τ(x ·), z

⟩L∞(G)∗,L∞(G)

(3.1)=(τ(y ·) ⊗ τ(x ·)

)◦ ∆(z)

= τ(y ·) ◦[(Id ⊗ τ)(1 ⊗ x)∆(z)

] (2.9)= τ(yR(·))

[(Id ⊗ τ)(∆(x)(1 ⊗ z))

]

=(τ(yR(·)) ⊗ τ(z ·)

)◦ ∆(x).

Theorem 3.3 Let G be a locally compact quantum group of Kac type. Suppose 1 6 p 6 ∞.We have a well-defined map completely contractive maps L1(G)⊗Lp(G) → Lp(G), f ⊗ g 7→ f ∗ gand Lp(G)⊗L1(G) → Lp(G), g ⊗ f 7→ g ∗ f .

Proof : Since the convolution ∗ of (3.1) induces a structure of matrix normed left L∞(G)∗-module on L∞(G)∗, the right dual action · of (3.3) induces a structure of matrix normedright L∞(G)∗-module on L∞(G). Moreover, the map R : L∞(G) → L∞(G) is an anti-∗-automorphism. For any [yij ] ∈ Mn(L1(G)) and any [xkl] ∈ Mm(L∞(G)), we deduce that

∥∥[yij ∗1,∞ xkl]∥∥

Mnm(L∞(G))

(3.6)=∥∥[xkl · τ(yijR(·))]

∥∥Mnm(L∞(G))

(2.38)

6∥∥[τ(yijR(·))]

∥∥Mn(L∞(G)∗)

∥∥[xkl]∥∥

Mm(L∞(G))=∥∥[yij ]

∥∥Mn(L1(G))

∥∥[xkl]∥∥

Mm(L∞(G)).

Recall that we have a completely contractive map ∆∗ : L∞(G)∗⊗L∞(G)∗ → L∞(G)∗, ω1 ⊗ω2 7→ω1∗ω2. By transport of structure, we deduce a completely contractive map ∆∗ : L1(G)op⊗L1(G)op →L1(G)op, f ⊗g 7→ f ∗1,1 g. Since we have (E⊗F )op = Eop⊗F op completely isometrically for anyoperator spaces E and F , we deduce a completely contractive map ∆∗ : L1(G)⊗L1(G) → L1(G),f ⊗ g 7→ f ∗1,1 g. By [ER, page 126], for any [yij ] ∈ Mn(L1(G)) and any [xkl] ∈ Mm(L1(G)) wededuce that ∥∥[yij ∗ xkl]

∥∥Mnm(L1(G))

6∥∥[yij ]

∥∥Mn(L1(G))

∥∥[xkl ]∥∥

Mm(L1(G)).

Recall that since Mn(Lp(G)) = (Mn(L∞(G)),Mn(L1(G))) 1p

by [Pis2, page 54]. So using Propo-

sition 3.2, we obtain by bilinear interpolation [Pis2, page 57] the inequality∥∥[yij ∗ xkl]

∥∥Mnm(Lp(G))

6∥∥[xkl ]

∥∥Mm(Lp(G))

∥∥[yij ]∥∥

Mn(L1(G)).

So by [ER, page 126] we have a well-defined map completely contractive map L1(G)⊗L∞(G) →L∞(G), x⊗ y 7→ x ∗ y.

The second part is similar and left to the reader.

Remark 3.4 It does not appear possible to drop the assumption “of Kac type” since it is statedin [LWW1] that the inequality ‖f ∗ g‖L∞(G) 6 ‖f‖L1(G) ‖g‖L∞(G) is not true for the compact

quantum group G = SUq(2) with q ∈] − 1, 1[−{0}. A slight generalization of the above proofcould maybe work for the more general case of locally compact groups whose scaling group istrivial.

13

Corollary 3.5 Let G be a locally compact quantum group of Kac type. Suppose 1 < p 6 ∞.Let f ∈ Lp(G). The map T : L1(G) → Lp(G), g 7→ f ∗ g is completely bounded with

(3.8) ‖T ‖cb,L1(G)→Lp(G) 6 ‖f‖Lp(G) .

Proof : By [ER, Proposition 7.1.2], we have an (complete) isometry

CB(Lp(G)⊗L1(G),Lp(G)) → CB(Lp(G),CB(L1(G),Lp(G)))

which maps the completely contractive map Lp(G)⊗L1(G) → Lp(G)), f ⊗ g 7→ f ∗ g to the mapLp(G) → CB(L1(G),Lp(G)), f 7→ (g 7→ f ∗ g). The conclusion is obvious.

Remark 3.6 If G is an abelian locally compact group and if 1 + 1r

= 1p

+ 1q, it is not difficult

to generalize the vector-valued Young’s inequality ‖f ∗ g‖Lr(R,X) 6 ‖f‖Lp(R) ‖g‖Lq(R,X) for the

group R from [ABHN1, Proposition 1.3.2] to the case of G. From this, it is obvious that if X is aBanach space any element f ∈ Lp(G) induces a convolution operator T : L1(G,X) → Lp(G,X),x 7→ f ∗ x satisfying

(3.9) ‖T ‖L1(G,X)→Lp(G,X) 6 ‖f‖Lp(G) .

Remark 3.7 We could probably easily extend the result to the case Lq(G)⊗Lp(G) → Lr(G)with some conditions on p, q and r. However, we does not have interesting application at thepresent moment.

3.2 Bounded versus completely bounded Fourier multipliers from L1(G)into Lp(G)

Let G be a locally compact quantum group of Kac type. Suppose 1 6 p < ∞. By Theorem3.3, we have a structure of L1(G)-bimodule on Lp(G) for any 1 6 p 6 ∞. So we can considerthe associated multipliers, see the the first paragraph of Section 2. We denote by M

1,p(G) thespace of multipliers from Lp(G) into Lq(G) what we call Fourier multipliers. In particular, aleft Fourier multiplier is a map T : L1(G) → Lp(G) satisfying

(3.10) T (x ∗ y) = T (x) ∗ y, x, y ∈ L1(G).

We denote by M1,pl (G) the space of left bounded Fourier multipliers.

Suppose 1 6 p 6 ∞. In accordance with discussion after Lemma 3.1, We will use thefollowing duality bracket

(3.11) 〈x, y〉Lp(G),Lp∗ (G)def= τ(xR(y)) = τ(yR(x)), x ∈ Lp(G), y ∈ Lp

∗

(G).

The reader should observe that this bracket is the best duality bracket in order to do harmonicanalysis on noncommutative Lp-spaces associated to compact quantum groups of Kac type.

We will use the following lemma which is a noncommutative analogue of [Dieu2, (14.10.9)].

Lemma 3.8 Let G be a locally compact quantum group of Kac type. Suppose 1 6 p < ∞. Ifx ∈ L1(G), y ∈ Lp

∗

(G) and z ∈ Lp(G), we have

(3.12) 〈z ∗ x, y〉Lp(G),Lp∗ (G) = 〈z, x ∗ y〉Lp(G),Lp∗ (G)

14

Proof : For any x ∈ L1(G) and any y ∈ L∞(G) ∩ Lp∗

(G), z ∈ L∞(G) ∩ Lp(G), we have

〈z ∗∞,1 x, y〉L∞(G),L1(G)(3.7)=⟨(

Id ⊗ τ(xR(·)))

◦ ∆(z), y⟩

L1(G),L∞(G)

= τ[((

Id ⊗ τ(xR(·)))

◦ ∆(z))R(y)

]=(τ(R(y)) ⊗ τ(xR(·))

)◦ ∆(z)

=(τ(y ·) ⊗ τ(x ·)

)◦ (R⊗R)∆(z) =

(τ(x ·) ⊗ τ(y ·)

)∆(R(z))

= τ[x(Id ⊗ τ)((1 ⊗ y)∆(R(z)))

]= τ

[xR((Id ⊗ τ)(∆(y)(1 ⊗R(z))

)]

= (τ(xR(·)) ⊗ τ)(∆(y)(1 ⊗R(z))

)= τ

[R(z)

(τ(xR(·)) ⊗ Id

)◦ ∆(y)

]

= 〈z,(τ(xR(·)) ⊗ Id

)◦ ∆(y)〉L1(G),L∞(G)

(3.7)= 〈z, x ∗1,∞ y〉Lp(G),Lp(G).

We conclude by density.

Now, we will show that the bounded Fourier multipliers from L1(G) into Lp(G) on a co-amenable locally compact quantum group are convolution operators. Recall that a locallycompact quantum groupG is co-amenable if and only if the Banach algebra L1(G) has a boundedapproximate unit with norm not more than 1. A known class of examples of coamenable locallycompact quantum groups is given by finite quantum groups.

Theorem 3.9 Let G be a co-amenable locally compact quantum group of Kac type. Suppose1 < p 6 ∞. Let T : L1(G) → Lp(G) be a map.

1. The operator T is a bounded left Fourier multiplier if and only if there exists f ∈ Lp(G)such that

(3.13) T (x) = f ∗ x, x ∈ L1(G).

2. The operator T is a bounded right Fourier multiplier if and only if there exists f ∈ Lp(G)such that

(3.14) T (x) = x ∗ f, x ∈ L1(G).

3. The map Lp(G) → M1,p(G), µ 7→ (f ∗ ·, · ∗ f) is a surjective isometry.

In the situation of the point or the point 2, f is unique, the map is completely bounded andwe have

(3.15) ‖f‖Lp(G) = ‖T ‖L1(G)→Lp(G) = ‖T ‖cb,L1(G)→Lp(G) .

Proof : 1. ⇐: Suppose that f belongs to Lp(G). The completely boundedness of T is aconsequence of Corollary 3.5 which in addition says that

(3.16) ‖T ‖cb,L1(G)→Lp(G)

(3.8)

6 ‖f‖Lp(G) .

Moreover, for any x, y ∈ L1(G), we have

T (x ∗ y)(3.13)

= f ∗ (x ∗ y) = (f ∗ x) ∗ y(3.13)

= T (x) ∗ y.

Hence (3.10) is satisfied, we conclude that T is a bounded left Fourier multiplier.

15

⇒: Suppose that T : L1(G) → Lp(G) is a bounded left Fourier multiplier. Since G is co-amenable, by [BeT1, Theorem 3.1] (and its proof) there exists a contractive approximative unit(bi) of the algebra L1(G). For any x ∈ L1(G), we have in L1(G)

(3.17) bi ∗ x −→ix.

For any i, we have

‖T (bi)‖Lp(G) 6 ‖T ‖L1(G)→Lp(G) ‖bi‖L1(G) 6 ‖T ‖L1(G)→Lp(G) .(3.18)

This implies that the net (T (bi)) lies in a norm bounded subset of the Banach space Lp(G).Note that if 1 < p < ∞, the Banach space Lp(G) is reflexive. By [Con1, Theorem 4.2 p. 132] ifp < ∞ and Alaoglu’s theorem if p = ∞ there exist a subnet (T (bj)) of (T (bi)) and f ∈ Lp(G)such that the net (T (bj)) converges to f for the weak topology of Lp(G) if p < ∞ and for theweak* topology of L∞(G) if p = ∞, i.e. for any z ∈ Lp

∗

(G) we have

(3.19)⟨T (bj), z

⟩Lp(G),Lp∗ (G)

−→j

〈f, z〉Lp(G),Lp∗ (G).

Lemma 3.10 For any x ∈ L1(G), we have in Lp(G)

(3.20) T (x) = limi

(T (bj) ∗ x

).

Proof : For any x ∈ L1(G) and any j, we have

‖T (x) − T (bj) ∗ x‖Lp(G)

(3.10)= ‖T (x) − T (bj ∗ x)‖Lp(G)

= ‖T (x− bj ∗ x)‖Lp(G) 6 ‖T ‖L1(G)→Lp(G) ‖x− bj ∗ x‖L1(G)

(3.17)−−−−→

j0.

For any x ∈ L1(G) any y ∈ Lp∗

(G), we have

⟨T (x), y

⟩Lp(G),Lp∗ (G)

(3.20)=

⟨limjT (bj) ∗ x, y

⟩Lp(G),Lp∗ (G)

= limj

⟨T (bj) ∗ x, y

⟩Lp(G),Lp∗ (G)

(3.12)= lim

j

⟨T (bj), x ∗ y

⟩Lp(G),Lp∗ (G)

(3.19)=

⟨f, x ∗ y

⟩Lp(G),Lp∗ (G)

(3.12)= 〈f ∗ x, y〉Lp(G),Lp∗ (G).

So we obtain (3.13).Using [Meg1, Theorem 2.5.21] if p < ∞ and [Meg1, Theorem 2.5.21] if p = ∞ in the first

inequality, we obtain

‖f‖Lp(G) =∥∥w∗ − lim

jT (bj)

∥∥Lp(G)

6 lim infj

‖T (bj)‖Lp(G)(3.21)

(3.18)

6 ‖T ‖L1(G)→Lp(G) 6 ‖T ‖cb,L1(G)→Lp(G) .

Combined with (3.16), we obtain (3.15). The uniqueness is essentially a consequence of thefaithfulness of the convolution, i.e. by [HNR1, Proposition 1].

2. The proof is similar to the proof of the point 1.

16

3. Let (L,R) be a bounded multiplier. Note that by [Daw1, Lemma 2.1], the mapR : L1(G) → Lp(G) is a bounded right Fourier multiplier. Then by the point 2, there existsf ∈ Lp(G) such that R(x) = x ∗ f . For any x, y ∈ L1(G), we have

(3.22) x ∗ L(y)(2.33)

= R(x) ∗ y = (x ∗ f) ∗ y = x ∗ (f ∗ y).

Since the convolution ∗ is faithful by [HNR1, Proposition 1], we deduce that for any y ∈ L1(G)we have L(y) = f ∗ y. Conversely, for any x, y ∈ L1(G), we have

x ∗ (f ∗ y) = (x ∗ f) ∗ y.

So by (2.33), the couple (f ∗ ·, · ∗ f) is a Fourier multiplier. The proof is complete.

Remark 3.11 The case p = 1 is described in [HNR2, Proposition 3.1] which says that if G isa co-amenable locally compact quantum group we have M

1,1(G) = M(G) where M(G) is themeasure algebra of G.

Remark 3.12 In the case of the group R, a similar result is stated by Hormander in theclassical paper [Hor1, Theorem 1.4]6.

Remark 3.13 By the proof of Corollary 3.5, the isometric map Φ: Lp(G) → CB(L1(G),Lp(G)),f 7→ f ∗ · is completely contractive. It could be possible that this map is even completely iso-metric, probably by generalizing the argument of (3.21). If p = 1, it is stated without proof in[HNR2, Proposition 3.1] for any co-amenable locally compact quantum group.

Remark 3.14 In the spirit of [HNR2, Proposition 3.1], we conjecture that if 1 < p 6 ∞ theco-amenability is necessary in order to have the isometry M

1,p(G) = Lp(G).

Now, we justify the name “Fourier multipliers” of our multipliers. Of course, we have asimilar result for right multipliers which is left to the reader.

Proposition 3.15 Let G be a co-amenable compact quantum group of Kac type. Suppose 1 <p 6 ∞. Let T : L1(G) → Lp(G) be a linear transformation. Then T is a bounded left Fouriermultiplier if and only if there exists a unique a ∈ ℓ∞(G) such that

(3.23) T (x)(π) = aπx(π), π ∈ Irr(G), x ∈ L1(G).

Proof : ⇒: By the first point of Theorem 3.9, there exists a unique f ∈ Lp(G) such that (3.13)is satisfied. Since the von Neuman algebra L∞(G) is finite we have the contractive inclusion

Lp(G) ⊂ L1(G). So note that by Proposition 2.1, the element adef= f belongs to ℓ∞(G).

Moreover, for any x ∈ L1(G) and any π ∈ Irr(G), we have

T (x)(π) = f ∗ x(π) = f(π)x(π) = aπx(π).

We have obtained (3.23).⇐: For any x, y ∈ L1(G) and any π ∈ Irr(G), we have

(3.24)

∧

T (x) ∗ y(π) = T (x)(π)y(π)(3.23)

= aπx(π)y(π)(2.17)

= aπx ∗ y(π)(3.23)

=

∧

T (x ∗ y)(π).

6. We reproduce the complete proof here: “follows from the fact that Lp′is the dual space of Lp when p < ∞”.

17

Hence

∧

T (x) ∗ y =

∧

T (x ∗ y). By Proposition 2.1, we infer that T (x) ∗ y = T (x ∗ y). By (3.10), we

deduce that T : L1(G) → Lp(G) is a left Fourier multiplier. It remains to show that this mapis bounded. Suppose that (xn) is a sequence of elements L1(G) converging to some elementx of L1(G) and that the sequence (T (xn)) converges to some y in Lp(G). Using Proposition2.1 in the second inequality and the last inequality (together with the contractive inclusionLp(G) ⊂ L1(G)), we obtain

∥∥T (x) − y∥∥ℓ∞(G)

6∥∥T (x) − T (xn)

∥∥ℓ∞(G)

+∥∥T (xn) − y

∥∥ℓ∞(G)

=∥∥∧

T (x− xn)∥∥ℓ∞(G)

+∥∥∧

T (xn) − y∥∥ℓ∞(G)

(3.23)

6∥∥a(x− xn)

∥∥ℓ∞(G)

+ ‖T (xn) − y‖L1(G)

6 ‖a‖ℓ∞(G)

∥∥x− xn∥∥ℓ∞(G)

+ ‖T (xn) − y‖Lp(G)

6 ‖a‖ℓ∞(G) ‖xn − x‖L1(G) + ‖T (xn) − y‖Lp(G) .

Passing to the limit, we infer that T (x) = y, hence T (x) = y by the injectivity of the Fouriertransform given by Proposition 2.1. By the closed graph theorem, we conclude that the operatorT : L1(G) → Lp(G) is bounded.

The uniqueness is left to the reader.

Remark 3.16 We could probably extend this result to the case of co-amenable locally compactquantum group of Kac type if 1 6 p 6 2 by using the Hausdorff-Young inequality [Coo1].

Definition 3.17 Let G be a co-amenable compact quantum group of Kac type. Suppose 1 <

p 6 ∞. Let T : L1(G) → Lp(G) be a bounded left Fourier multiplier. The unique a ∈ ℓ∞(G)

satisfying (3.23) is called the symbol of T . Moreover, we let Madef= T .

Let us state the particular case of co-commutative compact quantum groups. Here, we usethe classical notations, e.g. see [ArK] for background. Note that the case p = 1 is well-knownby the combination of [KaL1, Corollary 5.4.11. (ii)] and [ArK, Lemma 6.4].

Corollary 3.18 Let G be an amenable discrete group. Suppose 1 6 p 6 ∞. Any boundedFourier multiplier Mϕ : L1(VN(G)) → Lp(VN(G)) is completely bounded and we have

(3.25) ‖Mϕ‖cb,L1(VN(G))→Lp(VN(G)) = ‖Mϕ‖L1(VN(G))→Lp(VN(G)) .

Now, we return to the general case. A map T : Lp(G) → L∞(G) is a left Fourier multiplierif T (x ∗ y) = T (x) ∗ y for any x ∈ Lp(G) and any y ∈ L1(G). A simple duality argument givesthe following result.

Theorem 3.19 Let G be a co-amenable locally compact quantum group of Kac type. Suppose1 6 p < ∞.

1. A map T : Lp(G) → L∞(G) is a bounded left Fourier multiplier if and only if there existsf ∈ Lp

∗

(G) such that

(3.26) T (x) = f ∗ x, x ∈ Lp(G).

18

2. A map T : Lp(G) → L∞(G) is a bounded right Fourier multiplier if and only if there existsf ∈ Lp

∗

(G) such that

(3.27) T (x) = x ∗ f, x ∈ Lp(G).

In the situation of the point 1 or the point 2, f is unique and we have

(3.28) ‖f‖Lp∗ (G) = ‖T ‖cb,Lp(G)→L∞(G) = ‖T ‖Lp(G)→L∞(G) .

Proof : 1. ⇒: Suppose that there exists f ∈ Lp∗

(G) such that (3.26) is satisfied. By the point2 of Theorem 3.9, we have a completely bounded operator R : L1(G) → Lp

∗

(G), y 7→ y ∗ f . Forany x ∈ Lp(G) and any y ∈ L1(G), we have

⟨R(y), x

⟩Lp∗ (G),Lp(G)

= 〈y ∗ f, x〉Lp(G),Lp∗ (G)

(3.12)= 〈y, f ∗ x〉Lp(G),Lp∗ (G).

By [ER, Proposition 3.2.2], if p > 1 we infer that T : Lp(G) → L∞(G), x 7→ f ∗x is a well-definedcompletely bounded map satisfying (3.28). If p = 1, it is easy to adapt the argument.

⇐: Suppose that T : Lp(G) → L∞(G) is a bounded left Fourier multiplier. If p > 1, sinceLp(G) is reflexive, it is apparent e.g. from [Daw1, Lemma 10.1 (3)] that T is weak* continuous.So we can consider the preadjoint T∗ : L1(G) → Lp

∗

(G) on L1(G) of T (if p = 1 consider therestriction on L1(G) of the adjoint of T instead of T∗). For any x ∈ Lp(G) and any y, z ∈ L1(G)we have

⟨x, T∗(z ∗ y)

⟩Lp(G),Lp∗ (G)

=⟨T (x), z ∗ y

⟩L∞(G),L1(G)

(3.12)=

⟨T (x) ∗ z, y

⟩L∞(G),L1(G)

(2.35)=

⟨T (x ∗ z), y

⟩L∞(G),L1(G)

=⟨x ∗ z, T∗(y)

⟩Lp(G),Lp∗ (G)

(3.12)=

⟨x, z ∗ T∗(y)

⟩Lp(G),Lp∗ (G)

.

Hence for any y, z ∈ L1(G) we deduce that

T∗(z ∗ y) = z ∗ T∗(y).

Consequently by (2.36) the operator T∗ : L1(G) → Lp∗

(G) is a bounded right multiplier. Weinfer by the second point of Theorem 3.9 that there exists a unique f ∈ Lp

∗

(G) such thatT∗(y) = y ∗ f for any y ∈ L1(G). For any x ∈ Lp(G) and any y ∈ L1(G), we obtain

⟨T (x), y

⟩L∞(G),L1(G)

=⟨x, T∗(y)

⟩Lp(G),Lp∗ (G)

= 〈x, y ∗ f〉Lp(G),Lp∗ (G) = 〈f ∗ x, y〉L∞(G),L1(G).

Therefore T (x) = f ∗ x for any x ∈ Lp(G).2. The proof is similar to the proof of the point 1.The other assertions are very easy and left to the reader.

Definition 3.20 Let G be a co-amenable compact quantum group of Kac type. Suppose 1 6

p 6 ∞. Let T : Lp(G) → L∞(G) be a bounded left Fourier multiplier. We say that the element

adef= f of ℓ∞(G) is the symbol of T . We let Ma

def= T .

3.3 Another approach on completely bounded Fourier multipliers from

L1(G) into Lp(G)

In this section, we investigate the completely bounded Fourier multipliers from L1(G) into Lp(G)under weaker assumptions. We will use the following lemma which is an extension of [DJKRB,(3.20)] which states a similar result for positive elements of matrix spaces.

19

Lemma 3.21 Let M and N be von Neumann algebras equipped with normal semifinite faithfultraces. Suppose 1 6 p 6 ∞. For any positive element x of Lp(M,L1(N )), we have

(3.29) ‖x‖Lp(M,L1(N )) = ‖(Id ⊗ τ)(x)‖Lp(M)

where τ is the trace of N .

Proof : Here, we denotes the traces of M and N by the same letter τ . For any positivex ∈ Lp(M,L1(N )), we have using Hahn-Banach theorem with (2.6) in the last equality

‖x‖Lp(M,L1(N ))

(2.30)= sup

‖a‖2p∗ =1,a>0

∥∥(a⊗ 1)x(a⊗ 1)∥∥

L1(M⊗N )

= sup‖a‖2p∗ =1,a>0

(τ ⊗ τ)((a⊗ 1)x(a⊗ 1)

)= sup

‖a‖2p∗ =1,a>0

(τ ⊗ τ)((a2 ⊗ 1)x

)

= sup‖a‖2p∗ =1,a>0

τ[(Id ⊗ τ)

((a2 ⊗ 1)x

)]= sup

‖a2‖p∗ =1,a>0

τ[a2(Id ⊗ τ)(x)

]

= sup‖b‖

p∗ =1,b>0

τ[b(Id ⊗ τ)(x)

](2.6)=∥∥(Id ⊗ τ)(x)

∥∥Lp(M)

.

The following lemma is crucial here and in Section 3.5.

Lemma 3.22 Suppose 1 6 p 6 ∞. Let G be a compact quantum group of Kac type. Thecoproduct map induces an isometry ∆: Lp(G) → L∞(G,Lp(G)) and a contraction ∆: L1(G) →Lp(G,L1(G)).

Proof : Since ∆ is a trace preserving normal unital injective ∗-homomorphism, it induces a(complete) isometry ∆: Lp(G) → Lp(L∞(G)⊗L∞(G)) = Lp(G,Lp(G)). So for any x ∈ L∞(G),since the von Neumann algebra L∞(G) is finite, we have

(3.30) ‖∆(x)‖L∞(G,Lp(G)) > ‖∆(x)‖Lp(G,Lp(G)) = ‖x‖Lp(G) .

Now, we prove the reverse inequality. On the one hand, recall that by [BLM, Proposition 1.2.4]the ∗-homomorphism ∆: L∞(G) → L∞(G)⊗L∞(G) is (completely) contractive. On the otherhand, we can write in the case of a positive element x of L1(G)

‖∆(x)‖L∞(G,L1(G))

(3.29)=

∥∥(Id ⊗ τ)(∆(x))∥∥

L∞(G)

(2.14)= ‖τ(x)1‖L∞(G)(3.31)

= |τ(x)| ‖1‖L∞(G) = |τ(x)|(2.2)

6 ‖x‖L1(G) .

Now, consider an arbitrary element x of L1(G). By (2.4), there exists y, z ∈ L2(G) such thatx = yz and ‖x‖L1(G) = ‖y‖L2(G) ‖z‖L2(G). Note that a small approximation argument gives

∆(x) = ∆(yz) = ∆(y)∆(z). Moreover, we have

∥∥(Id ⊗ τ)(∆(y)∆(y)∗)∥∥ 1

2

L∞(G)

∥∥(Id ⊗ τ)(∆(z)∗∆(z))∥∥ 1

2

L∞(G)

(3.29)= ‖∆(y)∆(y)∗‖

12

L∞(G,L1(G)) ‖∆(z)∗∆(z)‖12

L∞(G,L1(G))

= ‖∆(yy∗)‖12

L∞(G,L1(G)) ‖∆(z∗z)‖12

L∞(G,L1(G))

(3.31)

6 ‖yy∗‖12

L1(G) ‖z∗z‖12

L1(G) = ‖y‖L2(G) ‖z‖L2(G) = ‖x‖L1(G) .

20

By (2.32), we deduce that ‖∆(x)‖L∞(G,L1(G)) 6 ‖x‖L1(G). So we have a contraction ∆: L1(G) →

L∞(G,L1(G)). We conclude by interpolation that ∆ induces a contraction ∆: Lp(G) →L∞(G,Lp(G)).

We know that ∆ induces an (complete) isometry ∆: L1(G) → L1(G,L1(G)). By interpola-tion, we deduce a contraction ∆: L1(G) → Lp(G,L1(G)).

Remark 3.23 Let G be a discrete group. It is possible to show that ∆ induces a contractivemap ∆: L2(VN(G)) → L∞(VN(G),L2(VN(G))) by the following argument. If a, b belongs tothe unit ball of L4(VN(G)) and if x =

∑s∈G αsλs is an element of Pol(G) with αs ∈ C, we

have using the orthonormality of the family (λs)s∈G and the argument of the proof of [Pis1,Proposition 3.9] in the fourth equality

∥∥(a⊗ 1)∆(x)(b ⊗ 1)∥∥

L2(VN(G)⊗VN(G))=

∥∥∥∥∥(a⊗ 1)∆

(∑

s∈G

αsλs

)(b⊗ 1)

∥∥∥∥∥L2(VN(G)⊗VN(G))

(2.19)=

∥∥∥∥∥(a⊗ 1)

(∑

s∈G

αsλs ⊗ λs

)(b⊗ 1)

∥∥∥∥∥L2(VN(G)⊗VN(G))

=

∥∥∥∥∥∑

s∈G

αsaλsb⊗ λs

∥∥∥∥∥L2(VN(G)⊗VN(G))

=

(∑

s∈G

α2s ‖aλsb‖

2L2(VN(G)

) 12 (2.3)

6

(∑

s∈G

α2s ‖a‖2

L4(VN(G) ‖λs‖2∞ ‖b‖2

L4(VN(G)

) 12

6

(∑

s∈G

α2s

) 12

‖a‖L4(VN(G) ‖b‖L4(VN(G) 6

∥∥∥∥∥∑

s∈G

αsλs

∥∥∥∥∥L2(VN(G))

= ‖x‖L2(VN(G)) .

Passing to the supremum on a and b, we obtain by (2.31)

‖∆(x)‖L∞(VN(G),L2(VN(G)) 6 ‖x‖L2(VN(G)) .

Now, we give a complement to Theorem 3.9.

Proposition 3.24 Let G be a compact quantum group of Kac type. Suppose 1 < p 6 ∞. Wesuppose that L∞(G) is hyperfinite if p < ∞. For any x ∈ Lp(G), we have

(3.32) ‖x ∗ ·‖cb,L1(G)→Lp(G) = ‖x‖Lp(G) .

Proof : If p = ∞, using [BLM, Proposition 1.2.4], we have a complete isometry

L∞(G) −→ L∞(G)⊗L∞(G)(2.22)−→ CB(L∞(G)∗,L

∞(G))

x 7−→ ∆(x) 7−→ (f 7→(f ⊗ Id

)(∆(x))

(3.5)= x ⋆ f

.

Recall that by (3.11) we have the completely isometric identification L1(G) → L∞(G)∗, y →τ(yR·)).

Suppose 1 < p < ∞. By Lemma 3.22, we have an isometry

Lp(G) −→ L∞(G,Lp(G))(2.28)

= L∞(G) ⊗min Lp(G)(2.21)−→ CB(L∞(G)∗,L

p(G))

x 7−→ ∆(x) 7−→ (f 7→(f ⊗ Id

)(∆(x))

(3.5)= x ⋆ f

which maps f on the convolution operator f ∗ ·.

21

Remark 3.25 Let G be a co-amenable compact quantum group of Kac type. Consider the iso-metric map Φ: Lp(G) → CB(L1(G),Lp(G)), f 7→ f ∗ · which is completely contractive. We willexplain here why the range Ran Φ of Φ is a completely contractively complemented subspace ofCB(L1(G),Lp(G)). Let E be the canonical trace preserving normal conditional expectation asso-ciated with the coproduct ∆: L∞(G) → L∞(G)⊗L∞(G). A completely contractive projectionfrom CB(L1(G),Lp(G)) onto Ran Φ is given by P : CB(L1(G),Lp(G)) → CB(L1(G),Lp(G)),T 7→ E(IdLp(G) ⊗T )∆ where ∆: L1(G) → Lp(G,L1(G)) and E : Lp(G,Lp(G)) → Lp(G). Indeed,note that these two last maps are completely contractive7. Now, it suffices to use [Arh1, Lemma2.1] and a variant of [Arh1, Proposition 3.5].

Remark 3.26 If G is a compact quantum group of Kac type such that L∞(G) is QWEP it ismaybe possible to adapt the proof of Proposition 3.24 in the case 1 < p < ∞. The difficultyis that the paper [Jun2] introduising vector-valued noncommutative Lp-spaces Lp(M, E) in thecase of a QWEP von Neumann algebra M is not definitively finished.

Remark 3.27 It is possible to give variants of Theorem 3.9 for the quantum torus Tdθ [CXY]or more generally for a twisted group von Neumann algebra VN(G, σ) by using the twistedcoproducts ∆σ,1 : VN(G, σ) → VN(G, σ)⊗VN(G), λσ,s 7→ λσ,s ⊗ λs and ∆1,σ : VN(G, σ) →VN(G)⊗VN(G, σ), λσ,s 7→ λs ⊗ λσ,s of [ArK, (4.5)] which is a unital injective normal ∗-homomorphism. Indeed, these coproducts induce by preduality an obvious structure of L1(VN(G))-bimodule on Lp(VN(G, σ)). So we can consider the Fourier multipliers from L1(VN(G, σ)) intoLp(VN(G, σ)). If VN(G, σ) is hyperfinite and if G is amenable, we obtain with the same methodan isometry M

1,pcb (G, σ) = Lp(VN(G)). In particular, for the quantum tori we have an isometry

M1,pcb (Tdθ) = Lp(T).

Remark 3.28 In a subsequent publication, we will give an example of bounded Fourier multi-plier on the free group Fn from L1(VN(Fn)) into Lp(VN(Fn)) which is not completely bounded.

3.4 Bounded versus completely bounded operators from L1(M) into

Lp(M)

In this section, we prove Theorem 3.31 that in general the bounded norm and the completelybounded norm are different for operator from L1(M) into Lp(M) where M is a semifinite vonNeumann algebra in sharp contrast with the case of Fourier multipliers on co-amenable compactquantum groups of Kac type. This result in the case p = ∞ can be seen as a von Neumannalgebra generalization of [Los1] (see also [KaL1, pp. 118-120]). See also [HuT1], [Osa1] and[Smi2] for strongly related papers.

Recall that if T : X → Y is a bounded operator between Banach spaces we have by [ER,(A.2.3) p. 333]:

(3.33) T is a quotient mapping ⇐⇒ T ∗ is an isometry.

Recall that a C∗-algebra A is n-subhomogeneous [BrO1, Definition 2.7.6] if all its irreduciblerepresentations have dimension at most n and subhomogeneous if A is n-subhomogeneousfor some n. It is known, e.g. [ShU1, Lemma 2.4], that a von Neumann algebra M is n-subhomogeneous if and only if there exist some distinct integers n1, . . . , nk 6 n, some abelianvon Neumann algebras Ai 6= {0} and a ∗-isomorphism

(3.34) M = Mn1 (A1) ⊕ · · · ⊕ Mnk(Ak).

7. It suffices by interpolation to prove that ∆: L1(G) → L∞(G, L1(G)) is completely contractive by a reiterationargument (repeat the proof of Lemma 3.22).

22

The following generalize [LeZ1, Lemma 2.3]. Note that the proof is simpler in the case whereM is finite since in this case we have an inclusion M ⊂ L1(M).

Proposition 3.29 Let M be a semifinite von Neumann algebra equipped with a normal semifi-nite faithful trace τ . Suppose that M is not n-subhomogeneous. There exists a non zero ∗-homomorphism γ : Mn+1 → M valued in M ∩ L1(M).

Proof : By [Tak1, Section V], we can consider the direct sum decomposition M = MI ⊕ MII

of M into a type I summand MI and a type II summand MII.

Assume that MII 6= {0}. According to [KaR2, Lemma 6.5.6], there exist n + 1 equivalentmutually orthogonal projections e1, . . . , en+1 in MII such that e1 + · · · + en+1 = 1. Then by[Tak1, Proposition V.1.22] and its proof, we have a ∗-isomorphism MII = Mn+1⊗(e1MIIe1).Let f be a non zero projection of e1MIIe1 with finite trace. For any positive matrix x ∈ Mn+1,we have

τMII(x⊗ f) = (Tr ⊗τe1MIIe1 )(x ⊗ f) = Tr(x)τe1MIIe1 (f) < ∞.

Hence the mapping γ : Mn+1 → MII ⊂ M, x 7→ x ⊗ f is a non zero ∗-homomorphism takingvalues in L1(M).

If MII = {0}, then M = MI is of type I. Since M is not subhomogeneous, it follows fromthe structure result (3.34) and [Tak1, Theorem V.1.27 p. 299] that there exists a Hilbert spaceH with dim(H) > n+1 and an abelian von Neumann algebra A such that M contains B(H)⊗Aas a summand. Let e ∈ B(H) be a rank one projection and define τA : A+ → [0,∞] by

τA(z)def= τM(e⊗ z).

Then τA is a normal semifinite faithful trace and τM coincides with Tr ⊗τA on B(H)+ ⊗ A+.Let f ∈ A be a non zero projection with finite trace. For any finite rank a ∈ B(H)+, it followsfrom above that

τM(a⊗ f) < ∞.

Now let (e1, . . . , en+1) be an orthonormal family in H . Then the mapping γ : Mn+1 → MII,

[aij ] →∑n+1

i,j=1 aijej ⊗ ei ⊗ f is a non zero ∗-homomorphism and the restriction of τM to the

positive part of its range is finite. Hence γ is valued in L1(M).

Below, ⊗ε denotes the injective tensor product of [Rya1, Chapter 3], ⊗ is the projectivetensor product of [Rya1, Chapter 2], ⊗ denotes the operator space projective tensor product.Moreover Cn and Rn denotes a Hilbert column space or a Hilbert row space of dimension n

defined in [BLM, 1.2.23]. For any z ∈ X ⊗ Y , we recall that

(3.35) ‖z‖X⊗εY

def= sup

{|〈z, x′ ⊗ y′〉| : ‖x′‖X∗ 6 1, ‖y′‖Y ∗ 6 1

}.

For any Banach spaces X and Y and any operator space E and F , we have isometries

(3.36) (X⊗Y )∗ = B(X,Y ∗) and (E⊗F )∗ = CB(E,F ∗).

The following lemma is stated in the case p = ∞ in [Los1] with a more complicated proof.

Lemma 3.30 Let n > 1 be an integer. Suppose 1 6 p 6 ∞. The canonical map Φn,p : Mn⊗min

Spn → Mn ⊗ε Spn, x⊗ y → x⊗ y is bijective and satisfies

∥∥Φ−1n,p

∥∥ > n1− 12p .

23

Proof : Note that using [Pis2, Corollary 2.7.7] in the second equality, we have isometrically

Cn ⊗min Cpn = Cn ⊗min (Cn,Rn) 1p

= (Cn ⊗min Cn,Cn ⊗min Rn) 1p

=(S2n, S

∞n

)1p

= S2pn .(3.37)

We deduce that∥∥∥∥∥n∑

i=1

ei1 ⊗ ei1

∥∥∥∥∥Mn⊗minS

pn

=

∥∥∥∥∥n∑

i=1

ei ⊗ ei

∥∥∥∥∥Cn⊗minCp

n

= ‖In‖S2pn

(3.37)= n1− 1

2p .

On the other hand, using the injectivity property [DeF, Proposition 4.3] of ⊗ε in the firstequality and the Cauchy-Schwarz inequality we have∥∥∥∥∥n∑

i=1

ei1 ⊗ ei1

∥∥∥∥∥Mn⊗εS

pn

=

∥∥∥∥∥n∑

i=1

ei ⊗ ei

∥∥∥∥∥ℓ2

n⊗εℓ2n

(3.35)= sup

{∣∣∣∣n∑

i=1

yizi

∣∣∣∣ : ‖y‖ℓ2n6 1, ‖z‖ℓ2

n6 1

}6 1.

With this lemma, we can prove the following result.

Theorem 3.31 Let M be a semifinite von Neumann algebra. Suppose 1 6 p 6 ∞. Thefollowing are equivalent.

1. The canonical map Θ: CB(L1(M),Lp(M)) → B(L1(M),Lp(M)) is an isomorphism.

2. M is subhomogeneous (in particular M is of type I).

In this case, if C is a positive constant such that ‖T ‖L1(M)→Lp(M) > C ‖T ‖cb,L1(M)→Lp(M) for

any completely bounded map T : L1(M) → Lp(M) then M is⌈( 1C

)2p

2p−1⌉-subhomogeneous.

Proof : Recall that we have canonical isometries

CB(L1(M),Lp(M))(3.36)

=(M∗⊗Lp

∗

(M))∗

and B(L1(M),Lp(M))(3.36)

=(M∗⊗Lp

∗

(M))∗.

1. ⇒ 2: Suppose that there exists a constant C > 0 such that

(3.38) ‖Θ(x)‖(M∗⊗Lp∗ (M))∗ > C ‖x‖(M∗⊗Lp∗ (M))∗ , x ∈ CB(L1(M),Lp(M)).

We define ndef=⌊( 1C

)2p

2p−1⌋

+ 1. We have ( 1C

)2p

2p−1 <⌊( 1C

)2p

2p−1⌋

+ 1 = n. Hence

(3.39)1

C< n

2p−12p = n1− 1

2p .

Suppose that M is not⌊( 1C

)2p

2p−1⌋-subhomogeneous. By Proposition 3.29, there exists a non

zero ∗-homomorphism γ : Mn → M taking values in M ∩ L1(M). Let τ ′ = τ ◦ γ : Mn → C.Then τ ′ is a non zero trace on the factor Mn hence there exists k > 0 such that τ ′ = kTr.

We deduce a complete isometry Jpdef= k− 1

p γ : Spn → Lp(M). Since Mn is finite-dimensional,the map J∞ is weak* continuous in the case p = ∞. The canonical embedding of J∞ ⊗Jp : Mn ⊗min S

pn → M ⊗min Lp(M) is an (complete) isometry. By (3.33), the preadjoint map

(Jp)∗ : Lp∗

(M) → Sp∗

n is a quotient map. By [Rya1, Proposition 2.5], we deduce that the tensorproduct (J∞)∗ ⊗(Jp)∗ : M∗⊗Lp

∗

(M) → S1n⊗Sp

∗

n is also a quotient map. From (3.33), we obtainthat the dual map

((J∞)∗ ⊗ (Jp)∗

)∗:(S1n⊗Sp

∗

n

)∗→(M∗⊗Lp

∗

(M))∗

24

is an isometry.Now, we will show that the following diagram commutes where j is defined by (2.21):

M ⊗min Lp(M) �� j // CB(L1(M),Lp(M)) �

� Θ //(M∗⊗Lp

∗

(M))∗

Mn ⊗min Spn

J∞⊗Jp

OO

Φn,p

//(S1n⊗Sp

∗

n

)∗= Mn ⊗ε S

pn

((J∞)∗⊗(Jp)∗)∗

OO

For any x ∈ Mn and any y ∈ Spn, we have

Θ ◦ j ◦ (J∞ ⊗ Jp)(x⊗ y) = Θ ◦ j(J∞(x) ⊗ Jp(y))

= Θ(〈J∞(x), ·〉M,L1(M)Jp(y)

)=⟨J∞(x), ·

⟩M,L1(M)

⊗⟨Jp(y), ·

⟩Lp(M),Lp∗ (M)

and

((J∞)∗ ⊗ (Jp)∗)∗Φn,p(x⊗ y) = ((J∞)∗ ⊗ (Jp)∗)∗(〈x, ·〉Mn,S1

n〈y, ·〉Mn,S1

n

)

=⟨x, (J∞)∗(·)

⟩Mn,S1

n

⊗⟨y, (Jp)∗(·)

⟩Mn,S1

n

.

For any z ∈ M ⊗ Lp(M), we deduce that

∥∥Φn,p(z)∥∥

Mn⊗εSpn

=∥∥((J∞)∗ ⊗ (Jp)∗

)∗Φn,p(z)

∥∥(M∗⊗Lp∗ (M))∗ =

∥∥Θ ◦ j(J∞ ⊗ Jp)(z)∥∥

(M∗⊗Lp∗ (M))∗

(3.38)

> C∥∥j(J∞ ⊗ Jp)(z)

∥∥M⊗minLp(M)

= C ‖z‖M⊗minLp(M) .

So ∥∥Φ−1n,p(z)

∥∥Mn⊗minS

pn6

1

C‖z‖Mn⊗εS

pn

We deduce that∥∥Φ−1

n,p

∥∥Mn⊗εS

pn→Mn⊗minS

pn6

1C

. By Lemma 3.30, we obtain n1− 12p 6

1C

. It is

impossible by (3.39).2. ⇒ 1: It is not difficult with (3.34) and left to the reader.

Remark 3.32 Let G be a discrete group. It is known by [Kan1] and [Smi1, Theorem 2] thatthe von Neumann algebra VN(G) is of type I if and only if G is virtually abelian, i.e. G hasan abelian subgroup of finite index. We deduce that if the von Neumann VN(G) satisfies theequivalent properties of Theorem 3.31, then G is virtually abelian. We will show the conversesomewhere and will examine the case of locally compact groups.

Remark 3.33 We can generalize almost verbatim the result to the case of maps from Lq(M)into Lp(M) using a folklore generalization of (3.37).

Remark 3.34 For the type III case, we can use a completely isometric version of [PiX, Theorem3.5] combined with Theorem 3.31 in the case of the unique separable hyperfinite factor of typeII1.

Corollary 3.35 Let M be a semifinite von Neumann algebra. Suppose 1 < p < ∞. Thefollowing are equivalent.

1. The canonical map CB(L1(M),Lp(M)) → B(L1(M),Lp(M)) is an isometry.

25

2. M is abelian.

Proof : 2. ⇒ 1.: Suppose that the von Neumann algebra M is abelian. In this case, we have by[Pis2, page 72] a complete isometry L1(M) = max L1(M). Then the point 1 is a consequenceof (2.23).

1. ⇒ 2.: This is a consequence of Theorem 3.31.

A direct consequence is the following corollary.

Corollary 3.36 Let G be a locally compact group. The following are equivalent.

1. We have a canonical isometry CB(L1(VN(G)),VN(G)) = B(L1(VN(G)),VN(G)).

2. The group G is abelian.

We close this Section by investigating the Schur multipliers from the Schatten trace classS1I into the space B(ℓ2

I) of bounded operators on the Hilbert space ℓ2I where I is an index set.

Proposition 3.37 Let I be an index set. Any A ∈ B(ℓ2I) induces a completely bounded Schur

multiplier MA : S1I → B(ℓ2

I) and we have

(3.40) ‖MA‖cb,S1I

→B(ℓ2I

) = ‖A‖B(ℓ2I

) .

Proof : Recall that we have a normal injective ∗-homomorphism ∆: B(ℓ2I) → B(ℓ2

I)⊗B(ℓ2I),

eij 7→ eij ⊗ eij . Using [BLM, Proposition 1.2.4], we obtain a complete isometry

Ψ: B(ℓ2I) −→ B(ℓ2

I)⊗B(ℓ2I)

(2.22)−→ CB(S1

I ,B(ℓ2I))

A 7−→ ∆(A) 7−→ (B 7→(〈·, B〉B(ℓ2

I),S1

I⊗ Id

)(∆(A))

.

For any i, j, k, l ∈ I, we have

Ψ(eij)(ekl)(2.22)

= 〈eij , ekl〉B(ℓ2I

),S1Ieij = δi=kδj=leij = Meij

(ekl).

Hence Ψ(eij) = Meij. It is not difficult to conclude that Ψ(A) = MA for any A ∈ B(ℓ2

I).

3.5 ℓp(Spd)-summing Fourier multipliers on L∞(G)

Suppose 1 6 p < ∞ and let d be an element of N ∪ {∞}. Let E and F be operator spaces andlet T : E → F be a linear map. Following [Pis1, page 58], we say that T is ℓp(Spd)-summing ifT induces a bounded map Idℓp(Sp

d) ⊗ T : ℓp(Spd) ⊗min E → ℓp(Spd(F )). In this case, we let

(3.41) ‖T ‖πp,d,E→F

def=∥∥Idℓp(Sp

d) ⊗ T

∥∥ℓp(Sp

d)⊗minE→ℓp(Sp

d(F ))

.

If d = ∞, it is not difficult to see that the ℓp(Sp)-summing maps coincide with the completelyp-summing maps8 and we have

(3.42) ‖T ‖πp,∞,E→F = ‖T ‖π◦p,E→F .

8. Recall that a map T : E → F is completely p-summing [Pis1, page 51] if T induces a bounded map IdSp ⊗T : Sp ⊗min E → Sp(F ).

26

Let T : E → F be a ℓp(Spd)-summing map. Since we have the isometric inclusion Spd(E) ⊂

Spd ⊗min E ⊂ ℓp(Spd) ⊗min E and the isometric inclusion S

pd(F ) ⊂ ℓp(Spd(F )), the map T is

bounded and

(3.43)∥∥IdSp

d⊗ T

∥∥S

p

d(E)→S

p

d(F )

6 ‖T ‖πp,d,E→F .

We need the following easy lemma.

Lemma 3.38 Suppose 1 6 p < ∞ and let d be an element of N ∪ {∞}. Let E,F,G,H beoperator spaces. Let T : E → F be a ℓp(Spd)-summing map. If the maps T1 : F → G andT2 : H → E are completely bounded then the map T1TT2 is ℓp(Spd)-summing and we have

(3.44) ‖T1TT2‖πp,d,H→G 6 ‖T1‖cb,F→G ‖T ‖πp,d,E→F ‖T2‖cb,H→E .

Proof : By [ER, Proposition 8.1.5], we have a well-defined (completely) bounded map Idℓp(Sp

d) ⊗

T2 : ℓp(Spd) ⊗min H → ℓp(Spd) ⊗min E with

(3.45)∥∥Idℓp(Sp

d) ⊗ T2

∥∥ℓp(Sp

d)⊗minH→ℓp(Sp

d)⊗minE

6 ‖T2‖cb,H→E .

Since T : E → F is a ℓp(Spd)-summing map, we have by definition a well-defined bounded mapIdℓp(Sp

d) ⊗ T : ℓp(Spd) ⊗min E → ℓp(Spd(F )). Moreover, we have

(3.46)∥∥Idℓp(Sp

d) ⊗ T1

∥∥ℓp(Sp

d(F ))→ℓp(Sp

d(G))

(2.27)

6 ‖T1‖cb,F→G .

We obtain the result by composition.

The following proposition gives a crucial example of ℓp(Spd)-summing map for the sequel.

Proposition 3.39 Suppose 1 6 p < ∞ and let d be an element of N ∪ {∞}. If M is ahyperfinite finite von Neumann algebra equipped with a normal finite faithful trace τ then thecanonical inclusion ip : M → Lp(M) is ℓp(Spd)-summing and

(3.47) ‖ip‖πp,d,M→Lp(M) = (τ(1))1p .

Proof : We can suppose that τ(1) = 1. Recall that if E is an operator space, we haveessentially9 by [Pis1, page 37] a contractive inclusion M ⊗min E → L1(M, E). Moreover, wehave Lp(M, E) = (M⊗minE,L

1(M, E)) 1p

by definition. By [Lun1, Proposition 2.4], we deduce

a contractive inclusion M ⊗min E → Lp(M, E). Taking E = ℓp(Spd) and using [Pis1, (3.6)] inthe last equality, we obtain a contractive inclusion

ℓp(Spd) ⊗min M ∼= M ⊗min ℓp(Spd) → Lp(M, ℓp(Spd)) ∼= ℓp(Spd(Lp(M)).

We conclude by (3.41) that the inclusion map ip : M → Lp(M) is ℓp(Spd)-summing with‖ip‖πp,d,M→Lp(M) 6 1. Note that ‖1‖Lp(M) = 1. Since

‖ip‖πp,d,M→Lp(M)

(3.43)

>∥∥IdSp

d⊗ ip

∥∥S

p

d(M)→S

p

d(Lp(M))

> ‖ip‖M→Lp(M) ,

we obtain the reverse inequality.

9. The result is stated with completely bounded instead of completely contractive.

27

Remark 3.40 If Fn is the free group with n generators where n > 2 is an integer then itis stated in [Jun3, Remark 3.2.2.5] that the canonical contractive inclusion ip : VN(Fn) →Lp(VN(Fn)) is not completely p-summing, i.e. is not ℓp(Sp)-summing. So the hyperfinitenessassumption in Proposition 3.39 cannot be removed. It would be interesting to characterizethe finite von Neumann algebras such that the canonical inclusion ip : M → Lp(M) is ℓp(Spd)-summing.

Proposition 3.41 Let G be a co-amenable compact quantum group of Kac type equipped witha normal finite faithful trace τ . Let a be an element of ℓ∞(G). Suppose 1 6 p < ∞ and let d bean element of N∪ {∞}. If a induces a completely bounded left Fourier multiplier Ma : Lp(G) →L∞(G) then a induces a ℓp(Spd)-summing left Fourier multiplier Ma : L∞(G) → L∞(G). In thiscase, we have

(3.48) ‖Ma‖πp,d,L∞(G)→L∞(G) 6 (τ(1))1p ‖Ma‖cb,Lp(G)→L∞(G) .

Proof : Suppose that a induces a completely bounded Fourier multiplier Ma : Lp(G) → L∞(G).Since G is co-amenable, by [BeT1, Theorem 3.1] the convolution algebra L1(G) has a boundedapproximate identity. By [Rua1, Theorem 4.5], we infer that the finite von Neumann algebraL∞(G) is hyperfinite. By Proposition 3.39, the map ip : L∞(G) −→ Lp(G) is ℓp(Spd)-summing

and the norm is ‖ip‖πp,d,L∞(G)→Lp(G) = (τ(1))1p . By considering the composition

L∞(G)ip−→ Lp(G)

Ma−−→ L∞(G)

we deduce by the ideal property (3.44) that Ma : L∞(G) → L∞(G) is ℓp(Spd)-summing and that

‖Ma‖πp,d,L∞(G)→L∞(G)

(3.44)

6 ‖Ma‖cb,Lp(G)→L∞(G) ‖ip‖πp,d,L∞(G)→Lp(G)

(3.47)= (τ(1))

1p ‖Ma‖cb,Lp(G)→L∞(G) .

Remark 3.42 If G is an abelian compact group G equipped with its normalized Haar measureand if ϕ : G → C is a complex function, we have the equalities(3.49)

‖Mϕ‖π◦

p ,L∞(G)→L∞(G) = ‖Mϕ‖

πp,L∞(G)→L∞(G) = ‖Mϕ‖Lp(G)→L∞(G) = ‖Mϕ‖cb,Lp(G)→L∞(G)

where ‖·‖πp,L∞(G)→L∞(G) is the classical p-summing norm. Indeed, the first equality is essen-

tially [JuP1, Remark 3.7] (see also [Pis1, Remark 5.13]). The third equality is (2.24) sinceL∞(G) = min L∞(G) as operator space by [Pis2, page 72]. The second equality is stated in[BeM1, Proposition 2]10 but the proof or the idea is not given. In respect to this beautifultradition of not to provide proofs, we does not give the argument here. However, we (really)confirm that a rather elementary proof exists11.

The following easy lemma is left to the reader.

Lemma 3.43 Let G be a co-amenable compact quantum group of Kac type. Suppose 1 < p 6

∞. Let T : L1(G) → Lp(G) be a completely bounded Fourier multiplier. Then

(3.50) ∆T = (Id ⊗ T )∆

10. The mathscinet review contains useful remarks.11. At least in the case b = ∞ of [BeM1, Proposition 2].

28

where we use ∆: Lp(G) → Lp(G⊗G) (on the left) and ∆: L1(G) → Lp(G,L1(G)) (on the right; see Lemma 3.22).

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace. Wedenote by Multa,b : M → Lp(M), x 7→ axb the multiplication map.

The following is a slight variation of [JuP1, Theorem 2.4], [JuP1, Theorem 2.4] and [JuP1,Remark 2.2]. We skip the details.

Theorem 3.44 Let M and N be hyperfinite von Neumann algebras such that M is equippedwith a normal finite faithful trace. Let T : M → N be a map. Then, the following assertionsare equivalent:

1. ‖T ‖πp,d,M→N 6 C.

2. There exist elements a, b ∈ M satisfying ‖a‖L2p(M) 6 1, ‖b‖L2p(M) 6 1 and a bounded

map T : Lp(M) → N such that

T = T ◦ Multa,b and∥∥T∥∥

Md(Lp(M))→Md(N )6 C.

Furthermore, ‖T ‖πp,d,M→N = inf{C : C satisfies any of the above conditions

}.

Now, we complete Propostion 3.41.

Theorem 3.45 Let G be a co-amenable compact quantum group of Kac type. We equip L∞(G)with its normalized normal finite faithful trace. Suppose 1 6 p < ∞. The element a of ℓ∞(G)induces a completely p-summing Fourier multiplier Ma : L∞(G) → L∞(G) if and only if itinduces a completely bounded Fourier multiplier Ma : Lp(G) → L∞(G). In this case, we have

(3.51) ‖Ma‖π◦p ,L

∞(G)→L∞(G) = ‖Ma‖cb,Lp(G)→L∞(G) = ‖Ma‖Lp(G)→L∞(G) .

Proof : ⇒: Suppose that we have a completely p-summing Fourier multiplier Ma1 : L∞(G) →L∞(G). By Theorem 3.44, there exist a, b ∈ L2p(G) satisfying ‖a‖L2p(G) 6 1 and ‖b‖L2p(G) 6 1

and a completely bounded map Ma1 : Lp(G) → L∞(G) such that

(3.52) Ma1 = Ma1 ◦ Multa,b and∥∥Ma1

∥∥cb,Lp(G)→L∞(G)

6 ‖Ma1‖π◦p ,L

∞(G)→L∞(G)

where Multa,b : L∞(G) → Lp(G). Let ∆: L∞(G) → L∞(G)⊗L∞(G) be the coproduct definedby (2.19). Let E : L∞(G)⊗L∞(G) → L∞(G) be the canonical trace preserving normal faith-ful conditional expectation associated with ∆. By using Lemma 3.22, we have a contraction∆: L1(G) → Lp

∗

(G,L1(G)). By duality, we deduce that we have a well-defined contractionE : Lp(G,L∞(G)) → L∞(G).

Since Ma1 : Lp(G) → L∞(G) is completely bounded, we deduce by (2.27) that the mapIdLp ⊗ Ma1 : Lp(G,Lp(G)) → Lp(G,L∞(G)) is completely bounded and that we have

∥∥IdLp ⊗ Ma1

∥∥cb,Lp(G,Lp(G))→Lp(G,L∞(G))

(2.27)

6∥∥Ma1

∥∥cb,Lp(G)→L∞(G)

(3.52)

6 ‖Ma1‖π◦p,L

∞(G)→L∞(G) .

(3.53)

By Lemma 3.22 applied with the opposite quantum group Gop [KuV1, Section 4], we havean isometry ∆op : L2(Gop) → L∞(Gop,L2(Gop)) = L∞(G,L2(G)), i.e. for any x ∈ L2(G) =L2(Gop) we have

‖∆op(x)‖L∞(G,L2(G)) = ‖x‖L2(G) .

29

By (2.30) applied with p = ∞ and q = r = 2, we deduce that for any elements c, d in the unitball of L4(G)

(3.54)∥∥(c⊗ 1)∆op(x)(d ⊗ 1)

∥∥L2(G⊗G)

6 ‖x‖L2(G) .

We infer that the map

(3.55) L2(G) → L2(G⊗G), x 7→ (c⊗ 1)∆op(x)(d ⊗ 1).

is a well-defined contraction. Since ∆ = Σ∆op, by Fubini’s theorem the map

(3.56) L2(G) → L2(G⊗G), x 7→ (1 ⊗ c)∆(x)(1 ⊗ d).

is also a contraction.

Proposition 3.46 Suppose 1 6 p 6 ∞. The map

(3.57) Θ: Lp(G) → Lp(G⊗G), x 7→ (1 ⊗ a)∆(x)(1 ⊗ b).

is a well-defined complete contraction.

Proof : Suppose 2 6 p 6 ∞. We can suppose that a and b have full supports. We will useStein interpolation [CwJ1] [Voi1, Theorem 2.1] [Lun1, Theorem 2.7]. For any complex numberz of the closed strip S = {z ∈ C : 0 6 Re z 6 1}, we consider the map

Tz : Pol(G) → L1(G), x 7→(1 ⊗ a

pz

2

)∆(x)

(1 ⊗ b

pz

2

)

where Pol(G) is the space of polynomials in L∞(G). Since a, b are elements of the unit ball ofL2p(G), the elements a

p

2 and bp

2 are elements of the unit ball of L4(G). For any t ∈ R and anyx ∈ Pol(G), we deduce that

‖T1+it(x)‖L2(G⊗G) =∥∥(1 ⊗ a

p+itp

2

)∆(x)

(1 ⊗ b

p+itp

2

)∥∥L2(G⊗G)

6∥∥(1 ⊗ a

p2

)∆(x)

(1 ⊗ b

p2

)∥∥L2(G⊗G)

(3.56)

6 ‖x‖L2(G) .

We infer that the map T1+it : L2(G) → L2(G⊗G) is contractive. Note that by [Pis2, page 139],the operator space L2(G) is an operator Hilbert space. We conclude that this map is evencompletely contractive by [Pis2, Proposition 7.2 (ii)].

For any t ∈ R and any x ∈ Pol(G), we have

‖Tit(x)‖L∞(G)⊗L∞(G) =∥∥(1 ⊗ a

itp

2

)∆(x)

(1 ⊗ b

itp

2

)∥∥L∞(G)⊗L∞(G)

= ‖∆(x)‖L∞(G)⊗L∞(G) .

We deduce that the map Tit : L∞(G) → L∞(G)⊗L∞(G) is completely contractive. Moreover,for any x ∈ Pol(G), the function S → L1(G), z 7→ Tz(x) is analytic on S and continuous andbounded on S. Finally, the functions R → L2(G⊗G), t 7→ T1+it(x) and R → L∞(G)⊗L∞(G),t 7→ Tit(x) are continuous. By Stein’s interpolation, we conclude by taking z = 2

pthat the map

(3.57) is a complete contraction.The case 1 6 p 6 2 is similar.

We have

∆Ma1

(3.50)= (Id ⊗Ma1)∆

(3.52)= (Id ⊗ Ma1Multa,b)∆

= (Id ⊗ Ma1)(Id ⊗ Multa,b)∆(3.57)

= (Id ⊗ Ma1)Θ.

30

Since E∆ = IdL∞(G), we deduce that Ma1 = E∆Ma1 = E(Id ⊗ Ma1)Θ. That means that wehave the following commutative diagram.

Lp(G,Lp(G))IdLp ⊗Ma1 // Lp(G,L∞(G))

E

��Lp(G)

Ma1

//

Θ

OO

L∞(G)

We conclude that a1 induces a bounded multiplier Ma1 : Lp(G) → L∞(G). Since G is co-amenable, by Theorem 3.19 this multiplier is completely bounded and we have

‖Ma1‖cb,Lp(G)→L∞(G) = ‖Ma1 ‖Lp(G)→L∞(G) =∥∥Θ(IdLp ⊗ Ma1)E

∥∥Lp(G)→L∞(G)

6 ‖Θ‖∥∥IdLp ⊗ Ma1

∥∥Lp(Lp)→Lp(L∞)

‖E‖(3.53)

6 ‖Ma1‖π◦p ,L

∞(G)→L∞(G) .

In the end of this section, we need the following which is a slight deepening of [Pis1, (1.7’)].

Lemma 3.47 Let E,F be operator spaces and T : E → F be a bounded map. If n > 1 is aninteger, we have

(3.58)∥∥IdSp

n⊗ T

∥∥S

pn(E)→S

pn(F )

= ‖IdMn⊗ T ‖Mn(E)→Mn(F ) .

Proof : Let x ∈ Mn(E). Consider some a ∈ S2pn , b ∈ S2p

n with ‖a‖S2pn

6 1, ‖b‖S2pn

6 1. Using[Pis1, (1.7)], in the last equality, we obtain

‖a · ((IdMn⊗ T )(x)) · b‖Sp

n(F ) =∥∥(IdSp

n⊗ T )(a · x · b)

∥∥S

pn(F )

6∥∥IdSp

n⊗ T

∥∥S

pn(E)→S

pn(F )

‖a · x · b‖Spn(E) 6

∥∥IdSpn

⊗ T∥∥Sp(E)→S

pn(F )

‖x‖Mn(E) .

Taking the supremum and using again [Pis1, (1.7)], we obtain

‖(IdMn⊗ T )(x)‖Mn(F ) 6

∥∥(IdSpn

⊗ T )∥∥S

pn(E)→S

pn(F )

‖x‖Mn(E) .

Hence ∥∥IdMn⊗ T

∥∥Mn(E)→Mn(F )

6∥∥IdSp

n⊗ T

∥∥S

pn(E)→S

pn(F )

.

Let x ∈ Spn(E). Let a ∈ S2pn , b ∈ S2p

n et y ∈ Mn(E) such that x = a · y · b. Using [Pis1, Lemma1.6 (ii)] in the first inequality, we obtain∥∥(IdSp

n⊗ T )(x)

∥∥S

pn(F )

=∥∥(IdSp

n⊗ T )(a · y · b)

∥∥S

pn(F )

= ‖a · (IdMn⊗ T )(y)) · b‖Sp

n(F )

6 ‖a‖S2pn

∥∥(IdMn⊗ T )(y)

∥∥Mn(F )

‖b‖S2pn

6 ‖IdMn⊗ T ‖Mn(E)→Mn(F ) ‖a‖S2p

n‖y‖Mn(E) ‖b‖S2p

n.

Passing to the infimum with [Pis1, Theorem 1.5], we obtain that∥∥(IdSp

n⊗ T )(x)

∥∥S

pn(F )

6∥∥IdMn

⊗ T∥∥

Mn(E)→Mn(F )‖x‖Sp

n(E) .

Finally ∥∥IdSpn

⊗ T∥∥S

pn(E)→S

pn(F )

6 ‖IdMn⊗ T ‖Mn(E)→Mn(F ) .

31

Remark 3.48 Let G be a finite quantum group. We equip L∞(G) with its normalized normal

finite faithful trace. We let dGdef= max

π∈Gdπ. Suppose 1 6 p < ∞ and d > dG. If a of ℓ∞(G)

we have

(3.59) ‖Ma‖πp,d,L∞(G)→L∞(G) = ‖Ma‖cb,Lp(G)→L∞(G) = ‖Ma‖Lp(G)→L∞(G) .

The proof is similar. We have a ∗-isomorphism L∞(G) = ⊕π∈G

Mdπ. The point is that we need

to control the norm of IdLp(G) ⊗ Ma1 by the ℓp(Spd)-summing norm of Ma1 . We have(3.60)

∥∥Ma1

∥∥S

p

d(Lp(G))→S

p

d(L∞(G))

(3.58)=

∥∥Ma1

∥∥Md(Lp(G))→Md(L∞(G))

6 ‖Ma1‖πp,d,L∞(G)→L∞(G) .

Moreover, we have

IdLp(G) ⊗ Ma1 = IdLp(⊕π∈G

Mdπ ) ⊗ Ma1 = Id⊕π∈G

Sp

dπ⊗ Ma1 = ⊕π∈G

(IdSp

dπ⊗ Ma1

).

If d = 1 < dG, we can prove the following result similarly with (2.25).

Theorem 3.49 Let G be a finite quantum group. We equip L∞(G) with its normalized normalfinite faithful trace. Suppose 1 6 p < ∞. If a ∈ ℓ∞(G) we have

(3.61)1

dG‖Ma‖Lp(G)→L∞(G) 6 ‖Ma‖πp,1,L∞(G)→L∞(G) 6 ‖Ma‖Lp(G)→L∞(G) .

4 Entropies and capacities

4.1 Definitions and transference

Generalizing the case of matrix algebras, it is natural to introduce the following definition.

Definition 4.1 Let M and N be finite von Neumann algebra equipped with normal finite faith-ful traces. Let T : L1(M) → L1(N ) be a quantum channel.

1. A value α > 0 is an achievable rate for classical information transmission through T if

(a) α = 0, or

(b) α > 0 and the following holds for every positive real number ε > 0: for all butfinitely many positive integers n and for m = ⌊αn⌋, there exist two quantum channelsΦ1 : ℓ1

2m → L1(M) and Φ2 : L1(N ) → ℓ12m such that

(4.1)∥∥Idℓ1

2m− Φ2T

⊗nΦ1

∥∥cb,ℓ1

2m →ℓ12m

< ε.

2. The classical capacity of T , denoted C(T ), is the supremum value of all achievable ratesfor classical information transmission through T .

3. Replacing ℓ12m by the Schatten space S1

2m , we define similarly the quantum capacity Q(T )of T .

Remark 4.2 Using the diagonal embedding J : ℓ12m → S1

2m and the canonical conditional ex-pectation E : S1

2m → ℓ12m , it is easy to check that

(4.2) Q(T ) 6 C(T ).

32

Remark 4.3 With obvious notations and standard arguments, we have

(4.3) C(T1T2) 6 min{

C(T1),C(T2)}

and Q(T1T2) 6 min{

Q(T1),Q(T2)}.

Moreover, we have

(4.4) C(T1 ⊗ T2) > C(T1) + C(T2) and Q(T1 ⊗ T2) > Q(T1) + Q(T2).

Generalizing again the case of matrix algebras by replacing the von Neumann entropy bythe Segal entropy, we equally introduce the following definition.

Definition 4.4 Let M be a finite von Neumann algebra equipped with a normal finite faithfultrace. Let T : L1(M) → L1(M) be a quantum channel. We define the Holevo capacity of T by

(4.5) χ(T )def= sup

{H

( N∑

i=1

λiT (ρi)

)−

N∑

i=1

λiH(T (ρi))

}

where the supremum is taken over all N ∈ N, all probability distributions (λ1, . . . , λN ) and allstates ρ1, . . . , ρN of L1(M),

Remark 4.5 A simple computation with (2.41) shows that the Holevo capacity χ(T ) is invari-ant if we replace the trace τ by the trace tτ where t > 0.

Remark 4.6 If τ is normalized, we have for allN ∈ N, all probability distributions (λ1, . . . , λN )and all states ρ1, . . . , ρN of L1(M)

H

( N∑

i=1

λiT (ρi)

)−

N∑

i=1

λiH(T (ρi)) 6 −N∑

i=1

λiH(T (ρi))(1.1)

6 −Hmin(T ).

Hence taking the supremum, we obtain the following observation

(4.6) χ(T ) 6 −Hmin(T ).

Remark 4.7 In [Arh4], we will prove that the classical capacity C(T ) of a quantum channelT : L1(M) → L1(M) is (1.6) generalizing the case of matrix algebras. The author did not try to

obtain a generalization of the formula Q(T ) = limn→+∞Q(1)(T⊗n)

n(which seems more delicate)

to von Neumann algebras.

Lemma 4.8 Let M and N be finite von Neumann algebra equipped with normal faithful finitetraces. Let J : M → N be a trace preserving normal injective ∗-monomorphism. Let x ∈ L1(M)be a positive element with ‖x‖1 = 1. If J1 : L1(M) → L1(N ) is the canonical extension of Jthen we have

(4.7) H(J1(x)) = H(x).

Proof : Suppose 1 < p < ∞. By [Arh3, Lemma 4.1], note that J induces an (not necessarilyonto) isometry Jp : Lp(M) → Lp(N ). Hence if x ∈ L1(M)+ with ‖x‖1 = 1, we have

(4.8) H(J1(x))(8.4)= −

d

dp‖Jp(x)‖pLp(N ) |p=1 = −

d

dp‖x‖pLp(M) |p=1

(8.4)= H(x).

33

Proposition 4.9 Let M and N be finite von Neumann algebra equipped with normal faithfulfinite traces. Suppose that J : M → N is a normal injective ∗-homomorphism and that E : N →M is the associated trace preserving normal conditional expectation. Let T : L1(M) → L1(M)be a quantum channel. We have(4.9)

Hmin(T ) = Hmin(J1TE1), χ(T ) = χ(J1TE1), C(T ) = C(J1TE1), Q(T ) = Q(J1TE1)

and if M,N are finite-dimensional

(4.10) Hcb,min(T ) = Hcb,min(J1TE1).

Proof : 1) We have

(4.11) Hmin(J1TE1)(1.2)= min

ρ>0,τ(ρ)=1H(J1TE1(ρ))

(4.7)= min

ρ>0,τ(ρ)=1H(TE1(ρ)).

Note that when ρ describe the states of L1(N ), the element E1(ρ) describe the states of L1(M).Hence, we conclude that

Hmin(J1TE1) = minρ′>0,τ(ρ′)=1

H(T (ρ′))(1.2)= Hmin(T ).

2) We have

χ(J1TE1)(1.5)= sup

{H

( N∑

i=1

λiJ1TE1(ρi)

)−

N∑

i=1

λiH(J1TE1(ρi))

}(4.12)

where the supremum is taken over all N ∈ N, all probability distributions (λ1, . . . , λN ) and allstates ρ1, . . . , ρN of L1(N ). Note that for any (λ1, . . . , λN ) and any states ρ1, . . . , ρN , we have

(4.13) H

( N∑

i=1

λiJ1TE1(ρi)

)= H

(J1

( N∑

i=1

λiTE1(ρi)

))(4.7)= H

( N∑

i=1

λiTE1(ρi)

)

and H(J1TE1(ρi))(4.7)= H(TE1(ρi)). So we have

(4.14) χ(J1TE1) = sup

{H

( N∑

i=1

λiTE1(ρi)

)−

N∑

i=1

λiH(TE1(ρi))

}

where the supremum is taken as before. Note that when ρ describe the states of L1(N ), theelement E1(ρ) describe the states of L1(M). Hence, we conclude that

χ(J1TE1) = sup

{H

( N∑

i=1

λiT (ρ′i)

)−

N∑

i=1

λiH(T (ρ′i))

}

where the supremum is taken over all N ∈ N, all probability distributions (λ1, . . . , λN ) and allstates ρ′

1, . . . , ρ′N of L1(M). Hence χ(J1TE1) = χ(T ).

3) Suppose that (4.1) is true with M = N . Since E1J1 = IdL1(M), we have E⊗n1 J⊗n

1 =IdL1(M⊗n). We infer that

∥∥Idℓ12m

− Φ2E⊗n1 (J1TE1)⊗nJ⊗n

1 Φ1

∥∥cb,ℓ1

2m →ℓ12m

=∥∥Idℓ1

2m− Φ2E

⊗n1 J⊗n

1 T⊗nE⊗n1 J⊗n

1 Φ1

∥∥cb,ℓ1

2m →ℓ12m

=∥∥Idℓ1

2m− Φ2T

⊗nΦ1

∥∥cb,ℓ1

2m →ℓ12m

< ε.

34

So α is an achievable rate for classical information transmission through J1TE1. Conversely, if∥∥Idℓ12m

− Φ2(J1TE1)⊗nΦ1

∥∥cb,ℓ1

2m →ℓ12m

< ε then

∥∥Idℓ12m

− Φ2J⊗n1 T⊗nE⊗n

1 Φ1

∥∥cb,ℓ1

2m →ℓ12m

< ε.

So using the encoding channel E⊗n1 Φ1 and the decoding channel Φ2J

⊗n1 , we see that α is an

achievable rate for classical information transmission through T .4) The proof is similar to the point 3.5) It is left to the reader.

4.2 Entropies and classical capacities of Fourier multipliers

Let M be a finite von Neumann algebra equipped with a normal finite faithful trace τ . Recallthat a quantum channel is a trace preserving completely positive map T : L1(M) → L1(M).By [SkV1, Theorem 2.1] if f ∈ L1(G) then the convolution map T : L1(G) → L1(G), x 7→ f ∗ xis a quantum channel if and and only if f is positive and ‖f‖L1(G) = 1.

If the trace τ is normalized, note that the numerical value S(µ)def= −τ(µ log µ) is negative.

Theorem 4.10 Let G be a finite quantum group equipped with a faithful trace τ . Let f bea element of L∞(G). Suppose that the convolution map T : L1(G) → L1(G), x 7→ f ∗ x is aquantum channel. Then

(4.15) Hcb,min(T ) = Hmin(T ) = −τ(f log f) = H(f).

Proof : Note that the compact quantum group G is of Kac type by [VDa2] and it is well-known that a finite quantum group is co-amenable. Hence, we can use Theorem 3.9. Using‖f‖L1(G) = 1 in the third equality, we obtain

Hcb,min(T )(1.4)= −

d

dp‖T ‖pcb,L1(G)→Lp(G) |p=1

(3.15)= −

d

dp‖f‖pLp(G) |p=1

(8.4)= −τ(f log f) = H(f).

By (1.3), the same computation is true without the subscript “cb”. So we equally haveHmin(T ) = H(f). The proof is complete.

Remark 4.11 In particular, Hmin is additive on convolution operators, i.e. Hmin(T1 ⊗ T2) =Hmin(T1) + Hmin(T2) if T1 and T2 are convolution operators.

Let n > 1 be an integer. Given a quantum channel T : S1n → S1

n, by [JuP1, pages 4-5] theHolevo bound is given by

(4.16) χ(T ) =d

dp‖T ∗‖πp∗,1,Mn→Mn

|p=1

where we refer to (1.5) for the definition of χ(T ).We extend this equality to maps acting on finite-dimensional vo Neumann algebras by using

the method of approximation of the proof of [ArK, Theorem 3.24].

Proposition 4.12 Let M = Mn1 ⊕ · · · ⊕ MnKbe a finite-dimensional von Neumann alge-

bra equipped with a faithful trace τ = λ1 Trn1 ⊕ · · · ⊕ λK TrnKwhere λ1, . . . , λK ∈ Q. Let

T : L1(M) → L1(M) be a quantum channel. Then we have

(4.17) χ(T ) =d

dp‖T ∗‖πp∗,1,M→M |p=1.

35

Proof : Case 1: All λk belong to N. Then let ndef=∑K

k=1 λknk. We consider the normal unitaltrace preserving ∗-homomorphism J : M → Mn defined by

J(x1 ⊕ · · · ⊕ xK) =

x1

. . .

x1 0. . .

xK

0. . .

xK

,

where xk appears λk times on the diagonal, k = 1, . . . ,K. Let E : Mn → M be the associatednormal faithful conditional expectation. Since EJ = IdM, we have T ∗ = EJT ∗EJ . Hence

‖T ∗‖πp,1,M→M =∥∥EJT ∗EJ

∥∥πp,1,M→M

(3.44)

6 ‖E‖cb ‖JT ∗E‖πp,1,Mn→Mn‖J‖cb 6 ‖JT ∗E‖πp,1,Mn→Mn

.

In a similar manner, we have

‖JT ∗E‖πp,1,Mn→Mn

(3.44)

6 ‖J‖cb ‖T ∗‖πp,1,Mn→Mn‖E‖cb 6 ‖T ∗‖πp,1,M→M .

Hence, we conclude that

(4.18) ‖T ∗‖πp,1,M→M = ‖JT ∗E‖πp,1,Mn→Mn.

Consequently, we have

d

dp‖T ∗‖πp∗,1,M→M |p=1

(4.18)=

d

dp‖JT ∗E‖πp∗,1,Mn→Mn

|p=1 =d

dp‖(J1TE1)∗‖πp∗,1,Mn→Mn

|p=1

(4.16)= χ(J1TE1)

(4.9)= χ(T ).

Case 2: All λk belong to Q+. Then there exists a common denominator of the λk’s, that means

that there exists t ∈ N such that λk =λ′

k

tfor some integers λ′

k. Note that the right-hand sideof (4.17) is invariant if we multiply the trace τ by t. Moreover, Remark (4.7) says that theleft-hand side of (4.17) is also invariant if we replace the trace τ by tτ . Thus, Case 1.2 followsfrom Case 1.1.

Remark 4.13 Undoubtedly, with a painful approximation argument in the spirit of the proofof [ArK, Theorem 3.24], we can extend the above result to the case of an arbitrary trace τ .

Now, we obtain a bound of the classical capacity of Fourier multipliers.

Theorem 4.14 Let G be a finite quantum group equipped with a normalized faithful trace. Letf be a element of L∞(G). Suppose that the convolution map T : L1(G) → L1(G), x 7→ f ∗ x isa quantum channel. Then

(4.19) C(T ) 6 −H(f) = τ(f log f)

36

Proof : 1. Note that the proof of Theorem 3.19 contains the description of the adjoint T ∗.Moreover, observe that for any element d of N ∪ {∞}, we have

limq→∞

‖T ∗‖πq,d,L∞(G)→L∞(G) = ‖IdMd⊗ T ∗‖Md(L∞(G))→Md(L∞(G)) = 1.

We let adef= f . We have

χ(T )(4.17)

=d

dp‖T ∗‖πp∗,1,L∞(G)→L∞(G) |p=1 =

d

dp‖f ∗ ·‖πp∗,1,L∞(G)→L∞(G) |p=1(4.20)

(2.42)= lim

q→+∞q ln ‖f ∗ ·‖πq,1,L∞(G)→L∞(G)

(3.48)

6 limq→+∞

q ln ‖Ma‖cb,Lq(G)→L∞(G)

=d

dp‖Ma‖cb,Lp∗ (G)→L∞(G) |p=1

(3.28)=

d

dp‖f‖Lp(G) |p=1

(8.6)= −H(f).

Another argument is the inequality χ(T )(4.6)

6 −Hmin(T )(4.15)

= −H(f).We can write L∞(G) = Mn1 ⊕ · · · ⊕ MnK

and its faithful trace satisfies the assumptionsof Proposition 4.17. Now, with the notations of the proof of Proposition 4.17, we have thefollowing “HSW formula”(4.21)

C(T )(4.9)= C(J1TE1)

(1.6)= lim

n→+∞

χ((J1TE1)⊗n)

n= limn→+∞

χ(J⊗n1 T⊗nE⊗n

1 )

n

(4.9)= lim

n→+∞

χ(T⊗n)

n.

By observing that the map f⊗n ∗ · : L1(Gn) → L1(Gn) is a convolution operator on the compact

quantum group Gn [Tro1, Section 2.5] whose the associated von Neumann algebra is L∞(Gn)def=

L∞(G)⊗ · · · ⊗L∞(G), we deduce that

C(T )(4.21)

= limn→+∞

χ(T⊗n)

n= lim

n→+∞

χ(f⊗n ∗ ·)

n

(4.20)

6 limn→+∞

−H(f⊗n)

n

(2.40)= lim

n→+∞

−nH(f)

n= −H(f).

Remark 4.15 Note that we can write −H(f) = τ(f log f) = S(τ(f ·), τ) where S(τ(f ·), τ) isthe relative entropy between the states τ(f ·) and τ . We deduce by [OhP1, Corollary 5.6] that−H(f) = 0 if and only if τ(f ·) = τ .

If f = 1, we have −H(f) = 0 so C(T ) = 0. It is not really surprising since for any state ϕand the Haar state τ we have τ ∗ ϕ = ϕ. So all the information of ϕ is lost by applying T .

Remark 4.16 Let G be a finite (classical) group and consider a measure of probability ν onG. It is elementary to check that the convolution operator T : ℓ1

G → ℓ1G, g 7→ ν ∗ g where

(ν ∗ g)(t)def=∑

s∈G

g(s−1t

)ν(s), t ∈ G,

is classical channel. By [CST1, page 104], the matrix of T is given12 by the Toeplitz matrix[ν(tr−1)]t,r∈G. Observe that the rows of such a matrix are permutations of each other. So such

12. For any t, r ∈ G, we have [T ]t,r = (ν ∗ er)(t) =∑

s∈Ger(s−1t)ν(s) = ν(tr−1).

37

a convolution operator T : ℓ1G → ℓ1

G is symmetric in the sense of [CoT1, page 190]. We deduceby [CoT1, (7.25)] that its classical capacity is given by

(4.22) C(T ) = log |G| − H(ν)

(and is achieved by a uniform distribution on the input) where H(ν) is the Shannon entropyof ν. If µG is the normalized Haar measure of G (i.e. the uniform probability measure on G)and if D denotes the relative entropy (Kullback-Leibler divergence) [CoT1, page 19], a simplecomputation left to the reader shows that

C(T ) = D(ν||µG).

Finally, we can also obtain the formula (4.22) by a combination of the formula C(T ) =d

dp ‖T ∗‖πp∗ ,ℓ∞G

→ℓ∞G

|p=1 of [JuP1, (1.2)]13 and Theorem 3.19. Indeed, first we consider the func-

tion f : G → C, s 7→ f(s) = |G|ν(s). Note that ‖f‖L1(G,µG) = 1|G|

∑s∈G f(s) = 1

|G|

∑s∈G ν(s) =

1. Now, we have14

C(T ) =d

dp‖T ∗‖πp∗ ,ℓ∞

G→ℓ∞

G|p=1 =

d

dp‖ν ∗ ·‖πp∗ ,ℓ∞

G→ℓ∞

G|p=1

=d

dp‖f ∗ ·‖πp∗ ,ℓ∞

G→ℓ∞

G|p=1

(3.49)=

d

dp‖f ∗ ·‖

ℓp∗

G→ℓ∞

G

|p=1

(3.28)=

d

dp‖f‖ℓp

G|p=1

(8.6)= −H(f, µG)

(2.41)= log |G| − H

(1

|G|f

)= log |G| − H(ν)

where we recall that H(·, µG) is the normalized entropy. So the bound of Theorem 4.14 is sharp.

5 Examples

5.1 Kac-Paljutkin quantum group

Now, we describe the Kac-Paljutkin quantum group KP introduced in [KaP1] (see also [FrG1]for some information). This is the smallest finite quantum group that is neither commutative

nor cocommutative. We consider the finite-dimensional von Neumann algebra L∞(KP)def=

13. If X and Y are Banach spaces, recall that here ‖·‖πp,X→Y denotes the p-summing norm πp of [DJT, page

31].

14. Recall that (f ∗ g)(t) =∫

Gg(s−1t)f(s) dµG(s) = 1

|G|

∑s∈G

g(s−1t)f(s).

38

C ⊕ C ⊕ C ⊕ C ⊕ M2(C). We will use the basis (e1, e2, e3, e4, e11, e12, e21, e22)15. The formulas

∆(e1) = e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 + e4 ⊗ e4 +1

2(e11 ⊗ e11 + e12 ⊗ e12 + e21 ⊗ e21 + e22 ⊗ e22)

∆(e2) = e1 ⊗ e2 + e2 ⊗ e1 + e3 ⊗ e4 + e4 ⊗ e3 +1

2(e11 ⊗ e22 + e22 ⊗ e11 + ie21 ⊗ e12 − ie12 ⊗ e21)

∆(e3) = e1 ⊗ e3 + e3 ⊗ e1 + e2 ⊗ e4 + e4 ⊗ e2 +1

2(e11 ⊗ e22 + e22 ⊗ e11 − ie21 ⊗ e12 + ie12 ⊗ e21)

∆(e4) = e1 ⊗ e4 + e4 ⊗ e1 + e2 ⊗ e3 + e3 ⊗ e2 +1

2(e11 ⊗ e11 + e22 ⊗ e22 − e12 ⊗ e12 − e21 ⊗ e21)

∆(e11) = e1 ⊗ e11 + e11 ⊗ e1 + e2 ⊗ e22 + e22 ⊗ e2 + e3 ⊗ e22 + e22 ⊗ e3 + e4 ⊗ e11 + e11 ⊗ e4

∆(e12) = e1 ⊗ e12 + e12 ⊗ e1 + ie2 ⊗ e21 − ie21 ⊗ e2 − ie3 ⊗ e21 + ie21 ⊗ e3 − e4 ⊗ e12 − e12 ⊗ e4

∆(e21) = e1 ⊗ e21 + e21 ⊗ e1 − ie2 ⊗ e12 + ie12 ⊗ e2 + ie3 ⊗ e12 − ie12 ⊗ e3 − e4 ⊗ e21 − e21 ⊗ e4

∆(e22) = e1 ⊗ e22 + e22 ⊗ e1 + e2 ⊗ e11 + e11 ⊗ e2 + e3 ⊗ e11 + e11 ⊗ e3 + e4 ⊗ e22 + e22 ⊗ e4

define the coproduct ∆: L∞(KP) → L∞(KP) ⊗ L∞(KP). The Haar state τ on L∞(KP) is givenby

(5.1) τ

(x1 ⊕ x2 ⊕ x3 ⊕ x3 ⊕ x4 ⊕

[a11 a12

a12 a22

])def=

1

8(x1 + x2 + x3 + x3 + x4 + 2a11 + 2a22).

The antipode R : L∞(KP) → L∞(KP) is given by

R(ek)def= ek, R(eij)

def= eji

where 1 6 k 6 4 and 1 6 i, j 6 2. We refer to [EnV1] for another construction of this finitequantum group.

A state on L∞(KP) identifies to a convex combination of four complex numbers and a stateof M2(C). Recall that by [Pet1, page 5] any state ψ of M2(C) can be written

ψ(A) = Tr(Aρ(x, y, z)

)where ρ(x, y, z) =

1

2

[1 + z x+ iyx− iy 1 − z

]

with A ∈ M2(C), x, y, z ∈ R and x2 + y2 + z2 6 1. So a state of L∞(KP) identifies to a sum

f = µ1 ⊕ µ2 ⊕ µ3 ⊕ µ4 ⊕µ5

2

[1 + z x+ iyx− iy 1 − z

]

where µ1, µ2, µ3, µ4, µ5 > 0, x, y, z ∈ R with µ1 + µ2 + µ3 + µ4 + µ5 = 1 and x2 + y2 + z2 6 1.The duality bracket (3.11) is given by

⟨x1 ⊕ x2 ⊕ x3 ⊕ x4 ⊕

[a11 a12

a21 a22

], µ1 ⊕ µ2 ⊕ µ3 ⊕ µ4 ⊕

µ5

2

[1 + z x+ iyx− iy 1 − z

]⟩

L∞(KP),L1(KP)

= x1µ1 + x2µ2 + x3µ3 + x4µ4 +µ5

2Tr

([a11 a12

a21 a22

] [1 + z x+ iyx− iy 1 − z

])

= x1µ1 + x2µ2 + x3µ3 + x4µ4 +µ5

2

[(1 + z)a11 + (x− iy)a21 + (x+ iy)a12 + (1 − z)a22

].

15. Here we have e3 = 0 ⊕ 0 ⊕ 1 ⊕ 0 ⊕

[0 00 0

]and e21 = 0 ⊕ 0 ⊕ 0 ⊕ 0 ⊕

[0 01 0

].

39

If f is a state, we have

(f ⊗ Id) ◦ ∆(e1) = (f ⊗ Id)

(e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 + e4 ⊗ e4 +

1

2(e11 ⊗ e11 + e12 ⊗ e12

+ e21 ⊗ e21 + e22 ⊗ e22)

)

= f(e1) ⊗ e1 + f(e2) ⊗ e2 + f(e3) ⊗ e3 + f(e4) ⊗ e4 +1

2

(f(e11) ⊗ e11 + f(e12) ⊗ e12

+ f(e21) ⊗ e21 + f(e22) ⊗ e22

)

= µ1e1 + µ2e2 + µ3e3 + µ4e4 +µ5

4

((1 + z)e11 + (x+ iy)e12 + (x − iy)e21 + (1 − z)e22

),

(f ⊗ Id) ◦ ∆(e2) = (f ⊗ Id)

(e1 ⊗ e2 + e2 ⊗ e1 + e3 ⊗ e4 + e4 ⊗ e3 +

1

2(e11 ⊗ e22 + e22 ⊗ e11

+ ie21 ⊗ e12 − ie12 ⊗ e21)

)

= f(e1) ⊗ e2 + f(e2) ⊗ e1 + f(e3) ⊗ e4 + f(e4) ⊗ e3 +1

2(f(e11) ⊗ e22 + f(e22) ⊗ e11

+ if(e21) ⊗ e12 − if(e12) ⊗ e21))

= µ1e2 + µ2e1 + µ3e4 + µ4e3 +µ5(1 + z)

4e22 +

µ5(1 − z)

4e11 + i

µ5(x− iy)

4e12 − i

µ5(x+ iy)

4e21

= µ2e1 + µ1e2 + µ4e3 + µ3e4 +µ5(1 − z)

4e11 + i

µ5(x− iy)

4e12 − i

µ5(x+ iy)

4e21 +

µ5(1 + z)

4e22,

(f ⊗ Id) ◦ ∆(e3) = (f ⊗ Id)

(e1 ⊗ e3 + e3 ⊗ e1 + e2 ⊗ e4 + e4 ⊗ e2 +

1

2(e11 ⊗ e22 + e22 ⊗ e11

− ie21 ⊗ e12 + ie12 ⊗ e21)

)

= f(e1) ⊗ e3 + f(e3) ⊗ e1 + f(e2) ⊗ e4 + f(e4) ⊗ e2 +1

2(f(e11) ⊗ e22 + f(e22) ⊗ e11

− if(e21) ⊗ e12 + if(e12) ⊗ e21))

= µ1e3 + µ3e1 + µ2e4 + µ4e2 +µ5(1 + z)

4e22 +

µ5(1 − z)

4e11 − i

µ5(x− iy)

4e12 + i

µ5(x+ iy)

4e21

= µ3e1 + µ4e2 + µ1e3 + µ2e4 +µ5(1 − z)

4e11 +

µ5(ix+ y)

4e12 +

µ5(ix− y)

4e21 +

µ5(1 + z)

4e22,

40

(f ⊗ Id) ◦ ∆(e4) = (f ⊗ Id)

(e1 ⊗ e4 + e4 ⊗ e1 + e2 ⊗ e3 + e3 ⊗ e2 +

1

2(e11 ⊗ e11 + e22 ⊗ e22

− e12 ⊗ e12 − e21 ⊗ e21)

)

= f(e1) ⊗ e4 + f(e4) ⊗ e1 + f(e2) ⊗ e3 + f(e3) ⊗ e2 +1

2(f(e11) ⊗ e11 + f(e22) ⊗ e22

− f(e12) ⊗ e12 − f(e21) ⊗ e21)

= µ1e4 + µ4e1 + µ2e3 + µ3e2 +µ5(1 + z)

4e11 +

µ5(1 − z)

4e22 −

µ5(x+ iy)

4e12 −

µ5(x− iy)

4e21

= µ4e1 + µ3e2 + µ2e3 + µ1e4 +µ5(1 + z)

4e11 −

µ5(x+ iy)

4e12 −

µ5(x− iy)

4e21 +

µ5(1 − z)

4e22,

(f ⊗ Id) ◦ ∆(e11) = (f ⊗ Id)(e1 ⊗ e11 + e11 ⊗ e1 + e2 ⊗ e22 + e22 ⊗ e2 + e3 ⊗ e22 + e22 ⊗ e3

+ e4 ⊗ e11 + e11 ⊗ e4

)

= f(e1) ⊗ e11 + f(e11) ⊗ e1 + f(e2) ⊗ e22 + f(e22) ⊗ e2 + f(e3) ⊗ e22 + f(e22) ⊗ e3

+ f(e4) ⊗ e11 + f(e11) ⊗ e4

)

= µ1e11 +µ5(1 + z)

2e1 + µ2e22 +

µ5(1 − z)

2e2 + µ3e22 +

µ5(1 − z)

2e3 + µ4e11 +

µ5(1 + z)

2e4

)

=µ5(1 + z)

2e1 +

µ5(1 − z)

2e2 +

µ5(1 − z)

2e3 + (µ1 + µ4)e11 + (µ2 + µ3)e22,

(f ⊗ Id) ◦ ∆(e12) = (f ⊗ Id)(e1 ⊗ e12 + e12 ⊗ e1 + ie2 ⊗ e21 − ie21 ⊗ e2 − ie3 ⊗ e21 + ie21 ⊗ e3

− e4 ⊗ e12 − e12 ⊗ e4

)

= f(e1) ⊗ e12 + f(e12) ⊗ e1 + if(e2) ⊗ e21 − if(e21) ⊗ e2 − if(e3) ⊗ e21 + if(e21) ⊗ e3

− f(e4) ⊗ e12 − f(e12) ⊗ e4

)

= µ1e12 +µ5(x + iy)

2e1 + iµ2e21 − i

µ5(x− iy)

2e2 − iµ3e21 + i

µ5(x− iy)

2e3 − µ4e12 −

µ5(x+ iy)

2e4

=µ5(x+ iy)

2e1 +

µ5(−ix− y)

2e2 +

µ5(ix+ y)

2e3 + (µ1 − µ4)e12 + (iµ2 − iµ3)e21,

(f ⊗ Id) ◦ ∆(e21) = (f ⊗ Id)(e1 ⊗ e21 + e21 ⊗ e1 − ie2 ⊗ e12 + ie12 ⊗ e2 + ie3 ⊗ e12 − ie12 ⊗ e3

− e4 ⊗ e21 − e21 ⊗ e4

)

= f(e1) ⊗ e21 + f(e21) ⊗ e1 − if(e2) ⊗ e12 + if(e12) ⊗ e2 + if(e3) ⊗ e12 − if(e12) ⊗ e3

− f(e4) ⊗ e21 − f(e21) ⊗ e4

)

= µ1e21 +µ5(x − iy)

2e1 − iµ2e12 + i

µ5(x+ iy)

2e2 + iµ3e12 − i

µ5(x+ iy)

2e3 − µ4e21 −

µ5(x− iy)

2e4

=µ5(x− iy)

2e1 +

µ5(ix − y)

2e2 +

µ5(−ix+ y)

2e3 +

µ5(−x+ iy)

2e4 + (iµ3 − iµ2)e12 + (µ1 − µ4)e21,

41

(f ⊗ Id) ◦ ∆(e22) = (f ⊗ Id)(e1 ⊗ e22 + e22 ⊗ e1 + e2 ⊗ e11 + e11 ⊗ e2 + e3 ⊗ e11 + e11 ⊗ e3

+ e4 ⊗ e22 + e22 ⊗ e4

)

= f(e1) ⊗ e22 + f(e22) ⊗ e1 + f(e2) ⊗ e11 + f(e11) ⊗ e2 + f(e3) ⊗ e11 + f(e11) ⊗ e3

+ f(e4) ⊗ e22 + f(e22) ⊗ e4

)

= µ1e22 +µ5(1 − z)

2e1 + µ2e11 +

µ5(1 + z)

2e2 + µ3e11 +

µ5(1 + z)

2e3 + µ4e22 +

µ5(1 − z)

2e4

=µ5(1 − z)

2e1 +

µ5(1 + z)

2e2 +

µ5(1 + z)

2e3 + (µ2 + µ3)e11 + (µ1 + µ4)e22.

With respect to the basis (e1, e2, e3, e4, e11, e12, e21, e22) of L∞(KP), we deduce that the matrixof the convolution operator T : L∞(KP) 7→ L∞(KP), y → f ∗ y is

µ1 µ2 µ3 µ41+z

2 µ5x+iy

2 µ5x−iy

2 µ51−z

2 µ5

µ2 µ1 µ4 µ31−z

2 µ5−x+iy

2 µ5x+iy

2 µ51+z

2 µ5

µ3 µ4 µ1 µ21−z

2 µ5ix+y

2 µ5 − −x+iy2 µ5

1+z2 µ5

µ4 µ3 µ2 µ11+z

2 µ5−x−iy

2 µ5−x+iy

2 µ51−z

2 µ51+z

4 µ51−z

4 µ51+z

4 µ51+z

4 µ5 µ1 − µ4 0 0 µ2 + µ3x+iy

4 µ5ix+y

4 µ5−ix−y

4 µ5−x−iy

4 µ5 0 µ1 − µ4 −iµ2 + iµ3 0x−iy

4 µ5ix−y

2 µ5ix−y

4 µ5−x+iy

4 µ5 0 iµ2 − iµ3 µ1 − µ4 01−z

4 µ51+z

4 µ51+z

4 µ51−z

2 µ5 µ2 + µ3 0 0 µ1 + µ4

.

If M8 is equipped with its normalized trace, we consider the normal unital trace preserving∗-homomorphism J : L∞(KP) → M8 defined by

J

(x1 ⊕ x2 ⊕ x3 ⊕ x3 ⊕ x4 ⊕

[a11 a12

a12 a22

])=

x1 0 0 0 0 0 0 00 x2 0 0 0 0 0 00 0 x3 0 0 0 0 00 0 0 x4 0 0 0 00 0 0 0 a11 a12 0 00 0 0 0 a12 a22 0 00 0 0 0 0 0 a11 a12

0 0 0 0 0 0 a21 a22

and the canonical trace preserving normal conditional expectation E : M8 → L∞(KP). Now wecan consider the channel JTE : M8 → M8 and combine the results of Section 4.1 and Section4.2.

Remark 5.1 A similar analysis could be done for the Sekine quantum groups [Sek1].

5.2 Dihedral groups

Recall that if G is a finite (classical) group, we have a ∗-isomorphism VN(G) → ⊕π∈GMdπ, λs 7→

⊕π∈Gπ(s). Note that the dihedral group D3 can be identified with the symmetric group S3 ={e, (123), (132), (12), (23), (13)}. Recall that S3 has three irreducible unitary representations: 1,sign and a two dimensional representation π : S3 → B(C2) defined by

π(e) =

[1 00 1

], π(123) =

[e

i2π3 0

0 e− i2π3

], π(132) =

[e− i2π

3 0

0 ei2π

3

]

42

and

π(12) =

[0 11 0

], π(23) =

[0 e− i2π

3

ei2π

3 0

], π(13) =

[0 e

i2π3

e− i2π3 0

].

In particular, we have a ∗-isomorphism Φ: VN(D3) 7→ C ⊕ C ⊕ M2(C) defined by

Φ(1) = 1 ⊕ 1 ⊕

[1 00 1

], Φ

(λ(12)

)= 1 ⊕ −1 ⊕

[0 11 0

],

Φ(λ(123)

)= 1 ⊕ 1 ⊕

[e

i2π3 0

0 e− i2π3

], Φ

(λ(23)

)= 1 ⊕ −1 ⊕

[0 e− i2π

3

ei2π

3 0

],

Φ(λ(132)

)= 1 ⊕ 1 ⊕

[e− i2π

3 0

0 ei2π

3

], Φ

(λ(13)

)= 1 ⊕ −1 ⊕

[0 e

i2π3

e− i2π3 0

].

If M6 is equipped with its normalized trace, we consider the normal unital trace preserving∗-homomorphism J : L∞(VN(D3)) → M6 defined by

J

(x1 ⊕ x2 ⊕

[a11 a12

a12 a22

])=

x1 0 0 0 0 00 x2 0 0 0 00 0 a11 a12 0 00 0 a12 a22 0 00 0 0 0 a11 a12

0 0 0 0 a21 a22

and the canonical trace preserving normal conditional expectation E : M6 → L∞(VN(D3). Now,if ϕ : D3 → C is a positive definite function with ϕ(e) = 1 we can consider the quantum channelJMϕE : M6 → M6 and combine the results of Section 4.1 and Section 4.2.

Remark 5.2 We have VN(D4) = C ⊕ C ⊕ C ⊕ C ⊕ M2(C).

6 The principle of transference in noncommutative anal-

ysis

6.1 Actions of locally compact quantum groups

The idea of this section is to start with a convolution operator Tf : L1(G) → L1(G), x 7→ f ∗ xon a quantum group G and to use a right action α of G on an algebra M (e.g. the matrixalgebra Mn) to construct a quantum channel Tf,α : M → M. These maps are related by theintertwining relation (6.5) and we can obtain sometimes some information on the channel Tf,αfrom the convolution operator by f . See the following subsection for illustrations. We willcontinue this section in another publication.

Let G be a locally compact quantum group. Recall the following notion of [Eno1, DefinitionI.1] and [Vae1, Definition 1.1]. We also refer to [DeC1] for more information on actions ofcompact quantum groups. A right action of G on a von Neumann algebra M is a normal unitalinjective ∗-homomorphism α : M → M⊗L∞(G) such that

(6.1) (α⊗ Id) ◦ α = (Id ⊗ ∆) ◦ α.

Similarly, a left action of G on a von Neumann algebra M is a normal unital injective ∗-homomorphism β : M → L∞(G)⊗M such that

(6.2) (Id ⊗ β) ◦ β = (∆ ⊗ Id) ◦ β.

43

Example 6.1 Suppose that G is a locally compact group acting on a von Neumann algebraM via a weak* continuous homomorphism β : G → Aut(M). We consider the locally compactquantum group (L∞(G),∆) where ∆: L∞(G) → L∞(G)⊗L∞(G) is defined in (2.18). The action

can be translated to a map β : M → L∞(G)⊗M = L∞(G,M), x 7→ β(x) where β(x)(s)def=

βs−1(x). Then it is known that α is an injective normal unital ∗-homomorphism satisfying (6.2).

If ω ∈ L∞(G)∗ and if α is a right action, we introduce the maps

(6.3) Tωdef= (Id ⊗ ω) ◦ ∆: L∞(G) → L∞(G) and Tω,α

def= (Id ⊗ ω) ◦ α : M → M

and if β is a left action

(6.4) Rωdef= (ω ⊗ Id) ◦ ∆: L∞(G) → L∞(G) and Rω,β

def= (ξ ⊗ Id) ◦ β : M → M.

The map Tω can be seen as a convolution map on the quantum group G. Note that the mapsTω,α and Rω,β are unital and completely positive. Now, we have the following relation betweenthese maps. If ω = τ(fR(·)), we also use the notation Tf,α

Proposition 6.2 Let G be a locally compact quantum group and M be a von Neumann algebra.Let α : M → M⊗L∞(G) be a right action and β : M → L∞(G)⊗M be a left action. Ifω ∈ L∞(G)∗, we have

(6.5) π ◦ Tω,α = (IdM ⊗ Tω) ◦ α and β ◦Rω,β = (Rω ⊗ IdM) ◦ β.

Proof : We have

(Id ⊗ Tω) ◦ α(6.3)= (Id ⊗ Id ⊗ ω)(Id ⊗ ∆) ◦ α

(6.1)= (Id ⊗ Id ⊗ ω)(π ⊗ Id) ◦ α

= (α⊗ ω) ◦ α = α ◦ (Id ⊗ ω) ◦ α(6.3)= α ◦ Tω,α.

and

(Rω ⊗ Id) ◦ β(6.4)= (ω ⊗ Id ⊗ Id)(∆ ⊗ Id) ◦ β

(6.2)= (ω ⊗ Id ⊗ Id)(Id ⊗ β) ◦ β

= (ω ⊗ β) ◦ β = β ◦ (ω ⊗ Id) ◦ β(6.4)= β ◦Rω,β .

Lemma 6.3 Let G be a locally compact quantum group of Kac type with normalized trace τand M be a von Neumann algebra equipped with a normal finite faithful trace τM. Let α : M →M⊗L∞(G) be a trace preserving right action. If ω ∈ L∞(G)∗ is a state then Tω,α : M → M istrace preserving, hence induces a quantum channel Tω,α : L1(M) → L1(M).

Proof : Using the preservation of the trace of the convolution operator Tω : L∞(G) → L∞(G)in the third equality, we obtain

τM ◦ Tω,α = (τM ⊗ τ) ◦ α ◦ Tω,α(6.5)= (τM ⊗ τ) ◦ (Id ⊗ Tω) ◦ α = (τM ⊗ τ) ◦ α = τM.

Of course, we have a similar result for trace preserving left actions.

44

The action α : M → M⊗L∞(G) by a compact quantum group G as in (6.1) is said to be

ergodic if the fixed point subalgebra Mα def= {x ∈ M : α(x) = x ⊗ 1} equals C1. Then M

admits a unique state hM determined by the following condition

(6.6) (Id ⊗ hG) ◦ α(x) = hM(x)1M, x ∈ M.

For an action of a classical group G as in Example 6.1, this equivalent to βs(x) = x for all s ∈ G

only if x is a scalar operator. If µG is the normalized Haar measure on G, then for each x ∈ Mit is knwon that the average

∫Gβs(x) dµG(s) is a scalar operator ω(x)1. Then ω is the unique

G-invariant state on M.If there is a continuous representation of a (classical) compact group G as an ergodic group

of ∗-automorphisms on a von Neumann algebra M then by [HLS1] M is finite and injectiveand the state hM is necessarily a trace.

A deeper analysis of the structure of ergodic actions of compact groups has been made byWassermann in [Was1], [Was2] and [Was3]. In particular, Wassermann remarkably showed thatthe compact group SU(2) only admits ergodic actions on von Neumann algebras of finite typeI.

The abstract theory of ergodic coactions of compact quantum groups on opertor algebrashas been initiated by Boca [Boc1]. The major difference with the compact group case is thatthe unique invariant state hM is not necessarily a trace. Indeed, Wang [Wan1] gave examplesof ergodic actions of universal unitary quantum groups on type III factors.

The interest of this definition lies in the following result and the next theorem.

Proposition 6.4 Let G be a compact group of Kac type equipped with normalized trace τ andM be a finite von Neumann algebra equipped with a normal finite faithful trace τM. Let α : M →M⊗L∞(G) be an ergodic right action. Suppose 1 6 p < ∞.

1. The map α induces a complete isometry α : Lp(M) → Lp(M,Lp(G)).

2. The map α induces an isometry α : L1(M) → Lp(M,L1(G)) which is completely contrac-tive.

Proof : 1. Since α is trace preserving, it suffices to use [Arh3, Lemma 4.1].2. On the one hand, we can write in the case of a positive element x of L1(M)

‖α(x)‖L∞(M,L1(G))

(3.29)=

∥∥(Id ⊗ hM)(α(x))∥∥

L∞(M)(6.7)

(6.6)=∥∥hM(x)1L∞(G)

∥∥L∞(M)

= |hM(x)| ‖1‖L∞(M) = |hM(x)| 6 ‖x‖L1(M) .

Now, consider an arbitrary element x of L1(M). By (2.4), there exists y, z ∈ L2(M) such thatx = yz and ‖x‖L1(M) = ‖y‖L2(M) ‖z‖L2(M). Note that α(x) = α(yz) = α(y)α(z). Moreover,we have

∥∥(Id ⊗ hM)(α(y)α(y)∗)∥∥

L∞(M)

∥∥(Id ⊗ hM)(α(z)∗α(z))∥∥

L∞(M)

(3.29)= ‖α(y)α(y)∗‖

12

L∞(M,L1(G)) ‖α(z)∗α(z)‖12

L∞(M,L1(G))

= ‖α(yy∗)‖12

L∞(M,L1(G)) ‖α(z∗z)‖12

L∞(M,L1(G))

(6.7)

6 ‖yy∗‖12

L1(M) ‖z∗z‖12

L1(M) = ‖y‖L2(M) ‖z‖L2(M) = ‖x‖L1(M) .

45

From (2.32), we infer that ‖α(x)‖L∞(G,L1(M)) 6 ‖x‖L1(M). We deduce a contraction α : L1(M) →

L∞(G,L1(M)). We conclude by interpolation with the isometry α : L1(M) → L1(G,L1(M))of the point 1 that α induces a contraction α : L1(M) → Lp(G,L1(M)). For the completecontractivity, use a reiteration argument with (2.32)instead of (2.4).

For the reverse inequality, it suffices to use the contractive inclusion Lp(G,L1(M)) ⊂L1(G,L1(M)) and the point 1.

Theorem 6.5 Let G be a co-amenable compact group of Kac type equipped with normalizedtrace τ and M be a finite von Neumann algebra equipped with a trace τM. Let α : M →M⊗L∞(G) be an ergodic right action. Then

(6.8) Hcb,min

(Tf,α

)> H(f) and χ(Tf,α) 6 −H(f).

Proof : Using the point 1 of Proposition 6.4 in the first equality and [Pis1, (3.1) page 39] inthe second inequality and the point 2 of Proposition 6.4 in last inequality, we have

‖Tf,α‖cb,L1(M)→Lp(M) = ‖α ◦ Tf,α‖cb,L1(M)→Lp(M,Lp(G))

(6.5)=∥∥(IdLp(M) ⊗ Tf

)◦ α∥∥

cb,L1(M)→Lp(M,Lp(G))

6∥∥IdLp(M) ⊗ Tf

∥∥cb,Lp(M,L1(G))→Lp(M,Lp(G))

‖α‖cb,L1(M)→Lp(M,L1(G))

6 ‖Tf‖cb,L1(G)→Lp(G) ‖α‖cb,L1(M)→Lp(M,L1(G))

(3.15)

6 ‖Tf‖cb,L1(G)→Lp(G) .

Moreover, ‖Tf,α‖cb,L1(M)→L1(M) = ‖Tf‖cb,L1(G)→L1(G) = 1. Now, we have

Hcb,min(Tf,α)(1.4)= −

d

dp‖Tf,α‖cb,L1(M)→Lp(M) |p=1

(2.42)= − lim

q→∞q ln ‖Tf,α‖cb,L1(M)→Lq∗ (M)

> − limq→∞

q ln ‖Tf‖cb,L1(G)→Lq∗ (G)

(2.42)= −

d

dp‖Tf‖Lp(G) |p=1

(1.4)= Hcb,min(Tf )

(4.15)= H(f).

The other inequality is a consequence of (4.6) and (8.14).

6.2 Transference I : from Fourier multipliers to Herz-Schur multipliers

Now, we show how our results on quantum channels on operator algebras allows us to recoverby transference some results of [GJL1] and [GJL2] for quantum channels T : S1

n → S1n acting

on the Schatten trace class space S1n which are Herz-Schur multipliers.

In this section, we use the notations of [ArK]. Here G is a finite group, Mϕ : VN(G) →VN(G), λs 7→ ϕ(s)λs is the Fourier multiplier on the von Neumann algebra VN(G) with symbolϕ : G → C and MHS

ϕ : B(ℓ2G) → B(ℓ2

G), x 7→ [ϕ(st−1)xst] is the Herz-Schur multiplier on B(ℓ2G)

with matrix Cϕdef= [ϕ(st−1)]. The operator MHS

ϕ is denoted Θ(ϕ) in the paper [CrN1].

Lemma 6.6 Let G be a finite group.

1. The map α : B(ℓ2G) → B(ℓ2

G)⊗VN(G), est 7→ est ⊗ λst−1 is well-defined trace preservingunital normal ∗-homomorphism.

2. If ∆ is defined in (2.19), we have (α⊗ Id) ◦ α = (Id ⊗ ∆) ◦ α, i.e. α is a right action onB(ℓ2

G).

46

3. Let ϕ : G → C be a complex function. If ϕ =∑

s∈G ϕsλs, we have Tϕ,α = MHSϕ and

Tϕ = Mϕ. In particular, we have

(6.9) α ◦MHSϕ =

(Id ⊗Mϕ

)◦ α.

4. Suppose 1 6 p < ∞. The map α induces a complete isometry α : SpG → SpG(Lp(VN(G)).

5. Suppose 1 6 p < ∞. The map α induces a complete contraction α : S1G → S

pG(L1(VN(G))).

Proof : 1. For any r, s, t, u ∈ G, we have on the one hand

α(ersetu) = δs=tα(eru) = δs=teru ⊗ λru−1 .

On the other hand, we have

α(ers)α(etu) = (ers ⊗ λrs−1 )(etu ⊗ λtu−1 ) = ersetu ⊗ λrs−1tu−1 = δs=teru ⊗ λru−1 .

Hence α is an homomorphism. Furthermore, for any r, s ∈ G, we have

(α(ers))∗ = (ers ⊗ λrs−1 )∗ = esr ⊗ λsr−1 = π(esr) = π(e∗

rs).

So the map α is a ∗-homomorphism. The normality is obvious by finite dimensionality. Finally,we have

α(1) = α

(∑

s∈G

ess

)=∑

s∈G

π(ess) =∑

s∈G

ess ⊗ λe = 1.

For any r, s ∈ G, we have(

Tr ⊗τG)(α(ers)

)=(

Tr ⊗τG)(ers ⊗ λrs−1 ) = Tr(ers)τG

(λrs−1

)= δr=s = Tr ers.

Hence α is trace preserving by linearity.2. For any s, t ∈ G, we have

(α⊗ Id) ◦ α(est) = (α⊗ Id)(est ⊗ λst−1 ) = α(est) ⊗ λst−1 = est ⊗ λst−1 ⊗ λst−1 .

and

(Id ⊗ ∆) ◦ α(est) = (Id ⊗ ∆)(est ⊗ λst−1 ) = est ⊗ ∆(λst−1 )(2.19)

= est ⊗ λst−1 ⊗ λst−1 .

3. For any s, t ∈ G, we have

Tϕ,α(est) = (Id ⊗ ϕ) ◦ α(est) = (Id ⊗ ϕ)(est ⊗ λst−1 )

= est ⊗ ϕ(λst−1 ) = ϕ(st−1)est = MHSϕ (est).

4. It suffices to use [Arh3, Lemma 4.1].5. Let E : B(ℓ2

G) → B(ℓ2G) be the trace preserving normal conditional expectation on the

diagonal. For any s, t ∈ G, note that

(6.10) (Id ⊗ τ)(α(est)) = (Id ⊗ τ)(est ⊗ λst−1 ) = τ(λst−1 )est = δs=test = E(est).

By linearity, we deduce that (IdS∞G

⊗ τ)α = E. In the case of a positive element x of S1n(S1

G),we can write

∥∥(IdS∞n

⊗ α)(x)∥∥S∞

n (S∞G

(L1(VN(G))))

(3.29)=

∥∥(IdS∞n (S∞

G) ⊗ τ)((IdS∞

n⊗ α)(x))

∥∥S∞

G(S∞

G)

(6.11)

=∥∥(IdS∞

n⊗ (Id ⊗ τ)α)(x)

∥∥S∞

G

(6.10)=

∥∥(IdS∞n

⊗ E)(x)∥∥S∞

G

6 ‖x‖S∞n (S∞

G) 6 ‖x‖S1

n(S1G

) .

47

Now, consider an arbitrary element x of S1n(S1

G). By (2.32) (2.4), there exists y, z ∈ S2n(S2

G)such that x = yz and ‖x‖S1

n(S1G

) = ‖y‖S2n(S2

G) ‖z‖S2

n(S2G

). Note that (Id ⊗α)(x) = (Id ⊗α)(yz) =

(Id ⊗ α)(y)(Id ⊗ α)(z). Moreover, we have

‖(Id ⊗ α)(y)((Id ⊗ α)(y))∗‖12

S∞n (S∞

G(L1(VN(G)))) ‖((Id ⊗ α)(z))∗(Id ⊗ α)(z)‖

12

S∞n (S∞

G(L1(VN(G))))

= ‖(Id ⊗ α)(yy∗)‖12

S∞n (S∞

G(L1(VN(G)))) ‖(Id ⊗ α)(z∗z)‖

12

S∞n (S∞

G(L1(VN(G))))

(6.11)

6 ‖yy∗‖12

S1n(S1

G)‖z∗z‖

12

S1n(S1

G)

= ‖y‖S2n(S2

G) ‖z‖S2

n(S2G

) = ‖x‖S1n(S1

G) .

From (2.32), we deduce that∥∥(IdS∞

n⊗ α)(x)

∥∥S∞

n (S∞G

(L1(VN(G))))6 ‖x‖S1

n(S1G

). So we have a

complete contraction α : S1G → S∞

G (L1(VN(G))). We conclude by interpolation with the com-plete isometry of the point 2 that α induces a complete contraction α : S1

G → SpG(L1(VN(G))).

Now, we obtain the following transference result.

Proposition 6.7 Let G be a finite group. Let ϕ : G → C be a complex function. We have

(6.12)∥∥MHS

ϕ

∥∥cb,S1

G→S

p

G

= ‖Mϕ‖cb,L1(VN(G))→Lp(VN(G)) .

Proof : Using the point 2 of Lemma 6.6 in the first equality and [Pis1, (3.1) page 39] in thesecond inequality and the point 3 of Lemma 6.6 in last inequality, we have

∥∥MHSϕ

∥∥cb,S1

G→S

p

G

=∥∥α ◦MHS

ϕ

∥∥cb,S1

G→S

p

G(Lp(VN(G))

(6.9)=∥∥(IdSp

G⊗Mϕ

)◦ α∥∥

cb,S1G

→Sp

G(Lp(VN(G))

6∥∥IdSp

G⊗Mϕ

∥∥cb,Sp

G(L1(VN(G)))→S

p

G(Lp(VN(G))

∥∥α∥∥

cb,S1G

→Sp

G(L1(VN(G)))

6∥∥Mϕ

∥∥cb,L1(VN(G))→Lp(VN(G))

∥∥α∥∥

cb,S1G

→Sp

G(L1(VN(G)))

6∥∥Mϕ

∥∥cb,L1(VN(G))→Lp(VN(G))

.

The reverse inequality is essentially a consequence of the construction of the embedding of [NeR,Section 3].

Remark 6.8 Of course, we can generalize some parts of the above results to the case of anarbitrary amenable discrete group G with a slightly more general proof.

If G is a locally compact group, recall that a function ϕ : G → C is positive definite if∑ni,j=1 αiαjϕ(s−1

j si) for any n ∈ N, any α1, . . . , αn ∈ C and any s1, . . . , sn ∈ G. If G isdiscrete, it is well-known that ϕ is positive definite if and only if

∑s∈G ϕ(s)λs is a positive

element of L1(VN(G)). If we have in addition ϕ(e) = 1 then∥∥∑

s∈G ϕ(s)λs∥∥

L1(VN(G))= 1.

Lemma 6.9 Let G be a finite group. Let ϕ : G → C be a positive definite function with ϕ(e) = 1.We have

(6.13) H

(∑

s∈G

ϕ(s)λs

)= H

(1

|G|Cϕ

)− log |G|.

Proof : Note that we have a ∗-homomorphism J : VN(G) → B(ℓ2G), λs 7→

∑t∈G est,t. Indeed,

for any s, r ∈ G, we have

J(λs)J(λr) =

(∑

t∈G

est,t

)(∑

u∈G

eru,u

)=∑

t,u∈G

est,teru,u =∑

u∈G

esru,u = J(λsr)

48

and

J(λs)∗ =

(∑

t∈G

est,t

)∗

=∑

t∈G

et,st =∑

t∈G

est,t = J(λs−1 ) = J(λ∗s).

Note that for any s ∈ G

TrJ(λs) = Tr

(∑

t∈G

est,t

)=∑

t∈G

Tr(est,t) =∑

t∈G

δst=t =∑

t∈G

δs=e = |G|τG(λs).

Hence Tr|G| ◦ Φ = τ , i.e. Φ preserves the normalized traces. We infer that

H

(∑

s∈G

ϕ(s)λs

)(4.7)= H

(J

(∑

s∈G

ϕ(s)λs

),

Tr

|G|

)= H

(∑

s∈G

ϕsJ(λs),Tr

|G|

)

= H

( ∑

s,t∈G

ϕsest,t,Tr

|G|

)= H

(Cϕ,

Tr

|G|

)(2.41)

= H

(1

|G|Cϕ

)− log |G|.

If ϕ : G → C is a positive definite function with ϕ(e) = 1, it is well-known (and obvious)that the Herz-Schur multiplier MHS

ϕ : B(ℓ2G) → B(ℓ2

G) is a quantum channel.

Corollary 6.10 Let G be a finite group. Let ϕ : G → C be a positive definite function withϕ(e) = 1. Then

(6.14) Hcb,min

(MHSϕ

)= H

(∑

s∈G

ϕ(s)λs

)= H

(1

|G|Cϕ

)− log |G|.

Proof : We have

Hcb,min

(MHSϕ

) (1.4)= −

d

dp

∥∥MHSϕ

∥∥pcb,S1

G→S

p

G

|p=1(6.12)

= −d

dp‖Mϕ‖pcb,L1(VN(G))→Lp(VN(G))

(1.4)= Hcb,min(Mϕ)

(4.15)= H

(∑

s∈G

ϕ(s)λs

).

Now, we can recover a result of [GJL1, Theorem III.3] and [GJL2, page 114] which gives thecompletely bounded entropy Hcb,min

(MHSϕ

)and the quantum capacity Q

(MHSϕ

)of a Herz-Schur

multiplier MHSϕ : S1

G → S1G where ϕ : G → C is a positive definite function with ϕ(e) = 1.

Corollary 6.11 Let G be a finite group. Let ϕ : G → C be a positive definite function withϕ(e) = 1. Then

(6.15) log |G| − H

(1

|G|Cϕ

)= −Hcb,min

(MHSϕ

)6 Q(1)

(MHSϕ

)= Q

(MHSϕ

)

Proof : Note that the quantum channel MHSϕ : B(ℓ2

G) → B(ℓ2G) is unital. The argument of

[GJL1, proof of Theorem III] gives log |G| − H(

1|G|Cϕ

) (6.14)= −Hcb,min

(MHSϕ

)6 Q(1)

(MHSϕ

).

Recall that Schur multipliers are degradable so Q(1)(MHSϕ

)= Q

(MHSϕ

).

49

6.3 Transference II: from Fourier multipliers to channels via repre-

sentations

Her, we connect our convolution operators with the quantum channels investigated in the paper[GJL1].

If u ∈ Mnu⊗L∞(G) is a unitary representation of a compact quantum group G (see (2.15)),

then it is folklore [Wan1, (2.10)] that the map α : Mnu→ Mnu

⊗ L∞(G), x 7→ u(x ⊗ 1)u∗ is aright action of G on the matrix algbra Mnu

in the sense of (6.1). It is easy to check that α isergodic if and only if u is irreducible. The next proposition allows us to recover the quantumchannel “Nf” of [GJL1, (9) page 10]. Here, we identify L1(G) with L∞(G)∗.

Proposition 6.12 Let G be finite quantum group and u be a unitary representation. Let f ∈L1(G) be a state. The map Tf,α : Mnu

→ Mnuis given by

(6.16) Tf,α(x) = (Id ⊗ τ)(u(x⊗ f)u∗

), x ∈ Mnu

.

Proof : By finite-dimensionality, we can write u =∑N

i=1 xi⊗ai where xi ∈ Mnuand ai ∈ L∞(G).

For any x ∈ Mnu, we have using the preservation of the trace τ by R in the last equality

Tτ(fR(·)),α(x)(6.4)= ((Id ⊗ τ(fR(·))) ◦ α(x) = ((Id ⊗ τ(fR(·)))(u(x ⊗ 1)u∗)

= ((Id ⊗ τ(fR(·)))

(( N∑

i=1

xi ⊗ ai

)(x⊗ 1)

( N∑

j=1

xj ⊗ aj

)∗)

=

N∑

i,j=1

((Id ⊗ τ(fR(·)))(xixx

∗j ⊗ a∗

i aj) =

N∑

i,j=1

(Id ⊗ τ(fR(·)))(xixx

∗j ⊗ a∗

i aj)

=

N∑

i,j=1

xixx∗j ⊗ τ

(fR(a∗

i aj))

=

N∑

i,j=1

xixx∗j ⊗ τ

(fR(aj)R(ai)

∗)

=N∑

i,j=1

xixx∗j ⊗ τ

(R(ai)

∗fR(aj))

= (Id ⊗ τ(R(·))

(( N∑

i=1

xi ⊗ ai

)(x⊗ f)

( N∑

j=1

xj ⊗ aj

)∗)

= (Id ⊗ τ(R(·)))(u(x⊗ f)u∗

)= (Id ⊗ τ)

(u(x⊗ f)u∗

).

So we can use Theorem 6.5.

7 Entanglement breaking and PPT Fourier multipliers

7.1 Multiplicative domains of Markov maps

Let A,B be C∗-algebras. Let T : A → B be a positive map. Recall that the left multiplicativedomain [Pau, page 38] of T is the set

(7.1) LMD(T )def={x ∈ A : T (xy) = T (x)T (y) for all y ∈ A

}.

By [Pau, Theorem 3.18] [Sto2, Definition 2.1.4], the right multiplicative domain RMD(T ) of acompletely positive map is16 a subalgebra of A. The right multiplicative domain17 is defined

16. The result remains true if T is a Schwarz map.17. We warn the reader that this notion is called multiplicative domain in [Sto2, Definition 2.1.4].

50

by

(7.2) RMD(T )def={x ∈ A : T (yx) = T (y)T (x) for all y ∈ A

}

We have RMD(T )∗ = LMD(T ). The multiplicative domain is MD(T )def= LMD(T ) ∩ RMD(T ).

If T : A → A is a map, we introduce the fixed point set

(7.3) Fix(T )def= {x ∈ A : T (x) = x}.

Recall that the definite set DT [Sto2, Definition 2.1.4] of a positive map T : A → A on aC∗-algebra A is defined by

(7.4) DTdef={x ∈ Msa : T (x2) = T (x)2

}.

Let (M, φ) and (N , ψ) be von Neumann algebras equipped with normal faithful states φand ψ, respectively. A linear map T : M → N is called a (φ, ψ)-Markov map if

(1) T is completely positive

(2) T is unital

(3) ψ ◦ T = φ

(4) for any t ∈ R we have T ◦σφt = σψt ◦T where (σφt )t∈R and (σψt )t∈R denote the automorphism

groups of the states φ and ψ, respectively.

When (M, φ) = (N , ψ), we say that T is a φ-Markov map. Note that a linear map T : M → Nsatisfying conditions (1) − (3) above is automatically normal. If, moreover, condition (4) issatisfied, then it is well-known, see [HaM, page 556] that there exists a unique completelypositive, unital map T ∗ : N → M such that

(7.5) φ(T ∗(y)x) = ψ(yT (x)), x ∈ M, y ∈ N .

It is easy to show that T ∗ is a (ψ, φ)-Markov map.The following observation is a generalization of [Rah1] (the result of [Rah1] was known

before) proved with a rather similar argument.

Lemma 7.1 Let M and N be von Neumann algebras equipped with normal faithful states φand ψ. Let T : M → N be a (φ, ψ)-Markov map. Then

LMD(T ) = RMD(T ) = MD(T ) = Fix(T ∗ ◦ T ).

Proof : Let x ∈ LMD(T ). Using the preservation of the states in the first equality, we have forany y ∈ M

φ(xy) = ψ(T (xy))(7.1)= ψ

(T (x)T (y)

) (7.5)= φ

(T ∗(T (x))y

).

Hence φ((x − T ∗(T (x)))y) = 0. Taking y = (x − T ∗(T (x)))∗, we obtain x = (T ∗ ◦ T )(x) byfaithfulness of φ. So x belongs to Fix(T ∗ ◦ T ).

Conversely, let x ∈ M such that T ∗ ◦ T (x) = x. Using the preservation of the states in thefirst equality and Schwarz inequality [Sto2, page 14], we obtain

φ(x∗x) = ψ(T (x∗x)) > ψ(T (x)∗T (x)

) (7.5)= φ

(T ∗(T (x)∗)x

)= φ

((T ∗(T (x)))∗x

)= φ(x∗x).

51

Since the extreme ends of the above equations are identical, the inequality is an equality. Hencewe have

ψ(T (x∗x)) = ψ(T (x)∗T (x)

).

By the faithfulness of the state ψ and Schwarz inequality, we infer that T (x∗x) = T (x)∗T (x).By [Pau, Theorem 3.18] [Sto2, Proposition 2.1.5], we conclude that x ∈ LMD(T ). The proof issimilar for RMD(T ). We also can use the equality RMD(T )∗ = LMD(T ).

Let G be a locally compact group. Suppose that ϕ : G → C is a positive definite functionwith ϕ(e) = 1. Note that the Fourier multiplier Mϕ : VN(G) → VN(G) is a τ -Markov mapwhere τ is the canonical normalized trace of the group von Neumann algebra VN(G). By

[HeR2, Corollary 32.7] [KaL1, Corollary 1.4.22], the set Gϕdef= {s ∈ G : |ϕ(s)| = 1} is a

subgroup of G and for any s ∈ Gϕ and any t ∈ G, we have

(7.6) ϕ(st) = ϕ(ts) = ϕ(t)ϕ(s).

In particular, ϕ is a character of the group Gϕ.The following result describes the multiplicative domains of Fourier multipliers which are

quantum channels.

Proposition 7.2 Let G be a discrete group. Let ϕ : G → C be a positive definite functionwith ϕ(e) = 1. An element

∑s xsλs of the group von Neumann algebra VN(G) belongs to the

multiplicative domain MD(Mϕ) of the Fourier multiplier Mϕ : VN(G) → VN(G) if and only iffor any s ∈ G the condition xs 6= 0 implies |ϕ(s)| = 1. That is

(7.7) MD(Mϕ) = VN(Gϕ).

Proof : It is easy to check18 that (Mϕ)∗ = Mϕ. By [HeR2, Theorem 32.4 (iii)], we obtain that(Mϕ)∗ = Mϕ. For any s ∈ G, we have

((Mϕ)∗ ◦Mϕ

)(λs) = (Mϕ ◦Mϕ)(λs) = ϕ(s)Mϕ(λs) = |ϕ(s)|2λs.

Hence the fixed points of the Fourier multiplier M∗ϕ ◦Mϕ are elements x =

∑s xsλs with non-

zero coefficients only at indices s where |ϕ(s)| = 1, that is s ∈ Gϕ. We conclude with Lemma7.1. The last formula is a consequence of [KaL1, Proposition 2.4.1].

7.2 PPT and entangling breaking maps

The first part of the following definition is a generalization of [HSR1, Theorem 4 (E)].

Definition 7.3 Let A,B be C∗-algebras. Let T : A → B be a linear map.

1. We say that T is entanglement breaking if for any C∗-algebra C and any positive mapP : B → C the map P ◦ T : A → C is completely positive.

2. We say that T is PPT if the maps T : A → B and T : A → Bop are completely positive.

Remark 7.4 An entanglement breaking map is PPT (in particular completely positive). Itsuffices to compose T with the identity IdB and the positive map B → Bop, x 7→ xop.

The following is a generalization of [RJV1, Lemma 3.1]. We give a simpler proof.

18. For any s, t ∈ G, we have τ(Mϕ(λs)λt) = ϕ(s)τ(λsλt) = ϕ(s)δs=t−1 and τ(λsMϕ(λt)) = ϕ(t−1)τ(λsλt) =ϕ(t−1)δs=t−1 .

52

Lemma 7.5 Let A,B be unital C∗-algebra. Let T : A → B be a unital completely positive mapsuch that the range Ran(T ) is contained in an abelian C∗-algebra B1, then T is entanglementbreaking.

Proof : Let P : B → C be a positive map where C is a C∗-algebra. Note that 1 belongsto Ran(T ), hence belongs to B1. The restriction P |B1 : B1 → C is positive hence completelypositive by [ER, Theorem 5.15]. We conclude that the composition P ◦T : A → C is completelypositive.

The following is a generalization of [RJV1, Lemma 3.1]. We replace the complicated com-putations of the proof by a simple conceptual argument which gives at the same time a moregeneral result. Recall that a positive map T is faithful [Sto2, page 10] if x > 0 and T (x) = 0imply x = 0.

Theorem 7.6 Let M be a von Neumann algebra equipped with a normal faithful state φ. LetT : M → M a unital PPT map. Then T (MD(T )) is an abelian algebra contained in the centerof the von Neumann algebra generated by the RanT . If T is in addition faithful, then MD(T )is abelian.

Proof : We have T (p)2 (7.1)= T (p) and T (p)∗ = T (p∗) = T (p). So T (p) is a projection. Note

that using p ∈ RMD(T ) in the last equality

T (p) ·op T (p) = T (p)T (p)(7.2)= T (p2).

So by (7.4), the selfadjoint element p belongs to the definite set DT of the map T : M →Mop. By [Sto2, Proposition 2.1.5], we deduce that p belongs to RMD(T : M → Mop) andto LMD(T : M → Mop) (since p is selfadjoint). Using p ∈ LMD(T ) in the first equality andp ∈ LMD(T : M → Mop) in the second equality, we infer that for any x ∈ M

(7.8) T (x)T (p) = T (xp) = T (x) ·op T (p) = T (p)T (x).

We let qdef= 1 − p. We deduce that for any x ∈ M

T (x) = T ((p+ q)x(p+ q)) = T (pxp) + T (qxp) + T (pxq) + T (qxp)

= T (pxp) + T (q)T (x)T (p) + T (p)T (x)T (q) + T (qxp)

= T (p)T (x)T (p) + T (q)T (x)T (q)

where we use T (p)T (q) = 0 = T (q)T (p) in the last equality. We deduce easily that

T (p)T (x) = T (x)T (p).

Hence each T (p) belongs to the commutant of RanT and obviously to RanT . We concludethat T (p) belongs to the center of the von Neumann generated by RanT .

If T is in addition faithful, restricting T on MD(T ) we obtain a ∗-isomorphism betweenMD(T ) and T (MD(T )). As T (MD(T )) is abelian, it follows that MD(T ) is also abelian.

The following is a generalization of [RJV1, Theorem 3.3]. As before, we simplify the proof.

Corollary 7.7 Let M be a von Neumann algebra equipped with a normal faithful state φ. LetE : M → M be a state preserving normal conditional expectation. The following properties areequivalent.

53

1. E is entangling breaking.

2. E is PPT.

3. the range RanE is abelian.

Proof : 1. ⇒ 2.: It is Remark 7.4.2. ⇒ 3.: Such a conditional expectation is faithful. It suffices to use Theorem 7.6. By [Str,

page 116], the range RanE of E is contained in MD(E) which is abelian.3. ⇒ 1.: It suffices to use Lemma 7.5.

Example 7.8 We deduce that the identity IdM : M → M is entangling breaking if and onlyif it is is PPT if and only if M is abelian. We recover in particular [Pau, Exercise 3.9 (vi)] forvon Neumann algebras.

Now, we give a necessary condition for the property PPT of a Fourier multiplier.

Theorem 7.9 Let G be a discrete group. Let ϕ : G → C be a function with ϕ(e) = 1. If theFourier multiplier Mϕ : VN(G) → VN(G) is a PPT quantum channel then the group Gϕ isabelian.

Proof : By Theorem 7.6, Mϕ(MD(Mϕ)) is an abelian algebra. By Proposition 7.2, we infer thatMϕ(VN(Gϕ)) is abelian and it is easy to check that Mϕ(VN(Gϕ)) = VN(Gϕ). We concludethat VN(Gϕ) is abelian, hence Gϕ also.

Example 7.10 Let G be discrete group and H be a subgroup of G. We can identify thevon Neumann algebra VN(H) as a subalgebra of VN(G). We can consider the canonical tracepreserving normal conditional expectation E : VN(G) → VN(G),

∑s∈G asλs 7→

∑s∈H asλs

onto VN(H). It is obvious that the map E is a Fourier multiplier with symbol 1H . We haveG1H

= {s ∈ G : |1H(s)| = 1} = H . We deduce with the results of this section that the quantumchannel E is PPT if and only if E is entangling breaking if and only if H is abelian.

8 Appendices

8.1 Appendix: output minimal entropy as derivative of operator norms

Here, we give a proof of (1.3) in the appendix for finite-dimensional von Neumann algebras, seeTheorem 8.4. This part does not aim at originality. Indeed, the combination of [ACN1] and[Aud1] gives a proof of this result for algebras of matrices. So, our result is a mild generalization.

Suppose 1 6 p, q 6 ∞. Let M,N be von Neumann algebras equipped with normal semifinitefaithful traces. Consider a positive linear map T : Lq(M) → Lp(N ). By an obvious generaliza-tion of the proof of [ArK, Proposition 2.18], such a map is bounded. We define the positive realnumber

(8.1) ‖T ‖+,Lq(M)→Lp(N )

def= sup

x>0,‖x‖q

=1

‖T (x)‖p .

We always have ‖T ‖+,Lq(M)→Lp(N ) 6 ‖T ‖Lq(M)→Lp(N ). We start to show that if T is in

addition 2-positive (e.g. completely positive) then we have an equality. See [Aud1] and [Sza1]for a version of this result for maps acting on Schatten spaces.

54

Proposition 8.1 Let M,N be von Neumann algebras equipped with normal semifinite faithfultraces. Suppose 1 6 p, q 6 ∞. Let T : Lq(M) → Lp(N ) be a 2-positive map. We have

(8.2) ‖T ‖Lq(M)→Lp(N ) = ‖T ‖+,Lq(M)→Lp(N ) .

Proof : Suppose ‖T ‖+,q→p 6 1. Let x ∈ Lq(M) with ‖x‖q = 1. By (2.4) (and a suitable

multiplication by a constant), there exist y, z ∈ L2q(M) such that

(8.3) x = y∗z with ‖y‖2q = ‖z‖2q = 1.

We have

[0 0y z

]∗ [0 0y z

]=

[0 y∗

0 z∗

] [0 0y z

]=

[y∗y y∗z

z∗y z∗z

]. We infer that

[y∗y y∗z

z∗y z∗z

]is a

positive element of Sq2(Lq(M)). Since T is 2-positive, the matrix (IdM2 ⊗ T )

([y∗y y∗z

z∗y z∗z

])=

[T (y∗y) T (y∗z)T (z∗y) T (z∗z)

]is positive. Moreover, we have ‖T (y∗y)‖p 6 ‖T ‖+,q→p ‖y∗y‖q 6 ‖y∗y‖q =

‖y‖22q

(8.3)= 1 and similarly ‖T (z∗z)‖p 6 1. Using [ArK, Lemma 2.13] in the first inequality, we

deduce that ‖T (x)‖p(8.3)= ‖T (y∗z)‖p 6

√‖T (y∗y)‖p ‖T (z∗z)‖p 6 1. Hence ‖T ‖q→p 6 1. By

homogeneity, we conclude that ‖T ‖q→p 6 ‖T ‖+,q→p. The reverse inequality is obvious.

The following proposition relying on Dini’s theorem is elementary and left to the reader.The first part is already present in [Seg1, page 625] without proof.

Proposition 8.2 Let M be a finite-dimensional von Neumann algebra equipped with a finitefaithful trace τ . For any positive element x ∈ M with ‖x‖1 = 1, we have

(8.4)d

dp‖x‖pLp(M) |p=1 = τ(x log x) = −H(x).

Moreover, the monotonically decreasing family of functions x 7→1−‖x‖p

Lp(M)

p−1 converge uniformly

to the function x 7→ −τ(x log x) = H(x) when p → 1+ on the subset of positive element x ∈ Mwith ‖x‖1 = 1.

Remark 8.3 It is not clear how generalize the second part of Proposition 8.2 to the case of aninfinite-dimensional finite von Neumann algebra M. Indeed L1(M) is a priori not a subset ofthe noncommutative Lp-space Lp(M) if p > 1.

Theorem 8.4 Let M,N be finite-dimensional von Neumann algebras equipped with finite faith-ful trace. Let T : M → N be a trace preserving 2-positive map. We have

(8.5) Hmin(T ) = −d

dp‖T ‖pL1(M)→Lp(N ) |p=1.

Proof : We can suppose that M and N are 6= {0}. Note that ‖T ‖L1(M)→L1(N ) 6 1 by a

classical argument [HJX, Theorem 5.1]. Since ‖T (1)‖1 = τN (T (1)) = τM(1) = ‖1‖1, we have‖T ‖L1(M)→L1(N ) = 1. Using the uniform convergence of Proposition 8.2 in the fifth equality

55

and the fact that T is trace preserving in the sixth equality, we obtain

−d

dp‖T ‖pL1(M)→Lp(N ) |p=1 = lim

p→1+

1 − ‖T ‖p1→p

p− 1

(8.2)= lim

p→1+

1 − ‖T ‖p+,1→p

p− 1

(8.1)= lim

p→1+

1 − maxx>0,‖x‖1=1 ‖T (x)‖pLp(N )

p− 1= lim

p→1+min

x>0,‖x‖1=1

1 − ‖T (x)‖pLp(N )

p− 1

(2.43)= min

x>0,‖x‖1=1limp→1+

1 − ‖T (x)‖pLp(N )

p− 1= min

x>0,‖x‖1=1

d

dp− ‖T (x)‖pLp(N ) |p=1

(8.4)= min

x>0,‖x‖1=1H(T (x))

(1.2)= Hmin(T ).

Remark 8.5 We also have the formulas Hmin(T ) = − ddp ‖T ‖L1(M)→Lp(N ) |p=1 and

(8.6)d

dp‖x‖Lp(M) |p=1 = τ(x log x) = −H(x).

for any positive element x ∈ M with ‖x‖1 = 1. Indeed, if f is a differentiable strictly positive

function with f(1) = 1 and if F (p) = f(p)1p , an elementary computation19 show that F ′(1) =

f ′(1).

Remark 8.6 With obvious notations, a standard argument gives

(8.7) Hmin(T1 ⊗ T2) 6 Hmin(T1) + Hmin(T2).

8.2 Appendix: multiplicativity of some completely bounded operator

norms

In this appendix, we prove in Theorem 8.10 the multiplicativity of the completely boundednorms ‖·‖cb,Lq→Lp if 1 6 q 6 p 6 ∞ and we deduce the additivity of Hcb,min(T ) in Theorem8.11. The multiplicativity result is stated in [DJKRB, Theorem 11] for Schatten spaces (whichimply the additivity result for matrix spaces). Unfortunately, the second part20 of the proof of[DJKRB, Theorem 11] is loosely false and consequently is not convincing. The only contributionof this section is to provide a correct proof. We also take this opportunity for generalize theresult to the more general setting of hyperfinite von Neumann algebras. However, it does notremove the merit of the authors of [DJKRB] to introduce this kind of questions. See also [Jen1]for a related paper.

The new ingredient is the following result which is crucial for the proof of Theorem 8.10.Note that this result is in the same spirit of shuffles of [EKR1, Theorem 4.1 and Corollary 4.2],[EfR1, Theorem 6.1]21 and [EfR2] which are well-known to the experts.

19. We have g′(p) = f(p)(

− 1p2 log f(p) + 1

p

f ′(p)f(p)

).

20. Note that the first part of the proof of [DJKRB, Theorem 11] is also incorrect but can be corrected withoutadditional ideas, see the proof of Theorem 8.10.21. If E1, E2, F1, F2 are operator spaces, we have a complete contraction (E1 ⊗eh F1) ⊗nuc (E2 ⊗eh F2) →(E1 ⊗nuc E2) ⊗eh (F1 ⊗nuc F2).

56

Proposition 8.7 Let M1, M2, M3 and M4 be hyperfinite von Neumann algebras equippedwith normal semifinite faithful trace. Suppose 1 6 q 6 p 6 ∞. The map

(8.8) Id ⊗ F ⊗ Id : Lp(M1,L

q(M2))

× Lp(M3,Lq(M4)

)→ Lp

(M1⊗M3,L

q(M2⊗M4))

which maps (x ⊗ y, z ⊗ t) on x⊗ z ⊗ y ⊗ t is a well-defined contraction.

Proof : Recall that by (2.20) we have an isometry M1 ⊗min L1(M2) −→ CBw∗ (M2,M1),a ⊗ b 7→ (z 7→ 〈b, z〉a) where the subscript w∗ means weak* continuous. Similarly, we havean isometry M3 ⊗min L1(M4) −→ CBw∗ (M4,M3) and (M1⊗M3) ⊗min L1(M2⊗M4) −→CBw∗ (M2⊗M4,M1⊗M3). If the tensors x ∈ M1 ⊗ L1(M2) and y ∈ M3 ⊗ L1(M4) iden-tify to the maps u : M2 → M1 and v : M4 → M3 then it is easy to check that the element(Id ⊗ F ⊗ Id)(x⊗ y) of (M1⊗M3) ⊗min L1(M2⊗M4) identify to the map u⊗ v : M2⊗M4 →M1⊗M3. By [BLM, page 40], we have

‖u⊗ v‖cb,M2⊗M4→M1⊗M36 ‖u‖cb,M2→M1

‖v‖cb,M4→M3.

We deduce that

∥∥(Id ⊗ F ⊗ Id)(x⊗ y)∥∥

(M1⊗M3)⊗minL1(M2⊗M4))6 ‖x‖M1⊗minL1(M2) ‖y‖M3⊗minL1(M4)) .

Hence by (2.28) the map L∞(M1,L1(M2))×L∞(M3,L

1(M4)) −→ L∞(M1⊗M3,L1(M2⊗M4))

is contractive. It is not difficult to prove that we have a contractive bilinear map

L∞(M1,M2) × L∞(M3,M4) −→ L∞(M1⊗M3,M2⊗M4).

By interpolating these two maps with [Pis1, (3.5)], we obtain a contractive bilinear map

L∞(M1,L

q(M2))

× L∞(M3,L

q(M4))

−→ L∞(M1⊗M3,L

q(M2⊗M4)).

Note that by [Pis1, (3.6)] we have a contractive bilinear map

Lq(M1,L

q(M2))

× Lq(M3,L

q(M4))

−→ Lq(M1⊗M3,L

q(M2⊗M4)).

Since 1p

=1− q

p

∞ +qp

q, we conclude by interpolation that the bilinear map (8.8) is contractive

Remark 8.8 The case p 6 q seems to us true but we does not need it. So we does not givethe proof.

Lemma 8.9 Let M1,M2,N1,N2 be hyperfinite von Neumann algebras. Suppose 1 6 q 6

p 6 ∞. Let T1 : Lq(M1) → Lp(N1) and T2 : Lq(M2) → Lp(N2) be linear maps. Then ifT1 ⊗T2 : Lq(Lq) → Lp(Lp) is completely bounded then T1 and T2 are completely bounded and wehave

‖T1 ⊗ T2‖cb,Lq(Lq)→Lp(Lp) > ‖T1‖cb,Lq→Lp ‖T2‖cb,Lq→Lp .

Proof : Let n,m > 1 be some integers. Consider some element x ∈ Spn(Lq(M1)) and y ∈Spm(Lq(M2)) such that

(8.9) ‖x‖Spn(Lq(M1)) 6 1 and ‖y‖Sp

m(Lq(M2)) 6 1.

57

Using the isometric flip Fp : Lp(M1⊗Mm) → Lp(Mm⊗M1) and the flip F : Lq(M1, Spm) →

Spm(Lq(M1)) and Proposition 8.7 in the third inequality, we have∥∥(IdSp

n⊗ T1)(x)

∥∥S

pn(Lp(M1))

∥∥(IdSpm

⊗ T2)(y)∥∥S

pm(Lp(M2))

=∥∥((IdSp

n⊗ T1)(x)

)⊗((IdSp

m⊗ T2)(y)

)∥∥Lp(Mn⊗M1⊗Mm⊗M2)

=∥∥(IdSp

n⊗ T1 ⊗ IdSp

m⊗ T2)(x ⊗ y)

∥∥Lp(Mn⊗M1⊗Mm⊗M2)

=∥∥(Id ⊗ Fp ⊗ Id)(IdSp

n⊗ T1 ⊗ IdSp

m⊗ T2)(x ⊗ y)

∥∥Lp(Mn⊗Mm⊗M1⊗M2)

=∥∥(IdSp

n⊗ IdSp

m⊗ T1 ⊗ T2)

((IdSp

n⊗ F ⊗ IdLq(M2))(x ⊗ y)

)∥∥Lp(Mn⊗Mm⊗M1⊗M2)

6∥∥IdSp

n⊗ IdSp

m⊗ T1 ⊗ T2

∥∥(p,p,q,q)→(p,p,p,p)

∥∥(IdSpn

⊗ F ⊗ IdLq(M2))(x ⊗ y)∥∥

(p,p,q,q)

6 ‖T1 ⊗ T2‖cb,Lq(Lq)→Lp(Lp)

∥∥(IdSpn

⊗ F ⊗ IdLq(M2))(x ⊗ y)∥∥

(p,p,q,q)

6 ‖T1 ⊗ T2‖cb,Lq(Lq)→Lp(Lp) ‖x‖Spn(Lq(M1)) ‖y‖Sp

m(Lq(M2))

(8.9)

6 ‖T1 ⊗ T2‖cb,Lq(Lq)→→Lp(Lp) .

Taking the supremum two times, we obtain∥∥IdSp

n⊗ T1

∥∥S

pn(Lq)→S

pn(Lp)

∥∥IdSpm

⊗ T2

∥∥S

pm(Lq)→S

pm(Lp)

6 ‖T1 ⊗ T2‖cb,Lq(Lq)→Lp(Lp) .

Taking again the supremum two times, we conclude by [Pis1, Corollary 1.2] that

‖T1‖cb,Lq→Lp ‖T2‖cb,Lq→Lp 6 ‖T1 ⊗ T2‖cb,Lq(Lq)→Lp(Lp) .

The main result of this section is the following theorem of multiplicativity.

Theorem 8.10 Let M1,M2,N1,N2 be hyperfinite von Neumann algebras equipped with nor-mal semifinite faithful trace. Suppose 1 6 q 6 p 6 ∞. Let T1 : Lq(M1) → Lp(N1) etT2 : Lq(M2) → Lp(N2) be completely bounded maps. Then the map T1 ⊗ T2 : Lq(M1 ⊗ N2) →Lp(N1 ⊗ N2) is completely bounded and we have

‖T1 ⊗ T2‖cb,Lq(Lq)→Lp(Lp) = ‖T1‖cb,Lq→Lp ‖T2‖cb,Lq→Lp .(8.10)

Proof : Using the completely contractive flip F : Lq(Lp) → Lp(Lq), x ⊗ y 7→ y ⊗ x, we haveusing [Pis1, (3.1)] in the last inequality

‖T1 ⊗ T2‖cb,Lq(Lq)→Lp(Lp) =∥∥(IdLp ⊗ T2)(T1 ⊗ IdLq )

∥∥cb,Lq(Lq)→Lp(Lp)

6∥∥IdLp ⊗ T2

∥∥cb,Lp(Lq)→Lp(Lp)

∥∥T1 ⊗ IdLq

∥∥cb,Lq(Lq)→Lp(Lq)

=∥∥IdLp ⊗ T2

∥∥cb,Lp(Lq)→Lp(Lp)

∥∥F(IdLq ⊗ T1)∥∥

cb,Lq(Lq)→Lp(Lq)

=∥∥IdLp ⊗ T2

∥∥cb,Lp(Lq)→Lp(Lp)

∥∥F∥∥

cb,Lq(Lp)→Lp(Lq)

∥∥IdLq ⊗ T1

∥∥cb,Lq(Lq)→Lq(Lp)

(2.29)

6∥∥IdLp ⊗ T2

∥∥cb,Lp(Lq)→Lp(Lp)

∥∥IdLq ⊗ T1

∥∥cb,Lq(Lq)→Lq(Lp)

6 ‖T1‖cb,Lq→Lp ‖T2‖cb,Lq→Lp .

The reverse inequality is Lemma 8.9.

We show how deduce the additivity of the completely bounded minimal entropy (to comparewith (8.7)). The proof follows the one of [DJKRB] for matrix spaces.

58

Theorem 8.11 Let M1,M2,N1,N2 be finite-dimensional von Neumann algebras equipped withfinite faithful trace. Let T1 : L1(M1) → L1(N1) et T2 : L1(M2) → L1(N2) be quantum channels.Then we have

(8.11) Hcb,min(T1 ⊗ T2) = Hcb,min(T1) + Hcb,min(T2).

Proof : For any p > 1, we have

1 − ‖T1 ⊗ T2‖pcb,L1(L1)→Lp(Lp)

p− 1

(8.10)=

1 − ‖T1‖pcb,L1→Lp ‖T2‖pcb,L1→Lp

p− 1

=1 − ‖T1‖pcb,L1→Lp + ‖T1‖pcb,L1→Lp − ‖T1‖pcb,L1→Lp ‖T2‖pcb,L1→Lp

p− 1

=1 − ‖T1‖pcb,L1→Lp

p− 1+ ‖T1‖pcb,L1→Lp

1 − ‖T2‖pcb,L1→Lp

p− 1.

We can suppose that the von Neumann algebras are 6= {0}. Note that similarly to the beginningof the proof of Theorem 8.4, we have ‖T1‖cb,L1→L1 = 1 and ‖T2‖cb,L1→L1 = 1. Taking the limit

when p → 1+ and using limp→1+ ‖T1‖pcb,L1→Lp = ‖T1‖cb,L1→L1 = 1, we obtain with (1.4) theadditivity formula (8.11).

Note that the derivative of (1.4) exists and there is a concrete description of the completelybounded minimal entropy Hcb,min(T ). The proof of the following result (which generalize [JuP2,Theorem 3.4]) is left to the reader.

Theorem 8.12 Let M be a finite-dimensional von Neumann algebra equipped with a faith-ful trace. Let T : L1(M) → L1(M) be a quantum channel. The function p 7→ ‖T ‖pcb,1→p isdifferentiable at p = 1 and the opposite of its derivative is given by

(8.12) Hcb,min(T ) = infρ∈L1(M)⊗L1(M)

{H[(Id ⊗ T )(ρ)

]− H

[(Id ⊗ τ)(ρ)

]}

where the infimum is taken on all states ρ. We can restrict the infimum to the pure states.

Remark 8.13 It should be not particularly difficult to give a second proof of the additivityof the completely bounded minimal entropy with the strong subadditivity of Segal’s entropy[Pod1] and the explicit formula (8.12).

With Remark 8.5, we also have the formula

(8.13) Hcb,min(T ) = −d

dp‖T ‖cb,L1(M)→Lp(M) |p=1.

Proposition 8.14 Let M be a finite-dimensional von Neumann algebra equipped with a faithfultrace. Let T : L1(M) → L1(M) be a quantum channel.

(8.14) Hcb,min(T ) 6 Hmin(T )

Proof : We have

Hcb,min(T )(8.13)

= −d

dp‖T ‖cb,L1(M)→Lp(M) |p=1

(2.42)= − lim

q→∞q ln ‖T ‖cb,L1(M)→Lq∗ (M)

6 − limq→∞

q ln ‖T ‖L1(M)→Lq∗ (M)

(2.42)= −

d

dp‖T ‖L1(M)→Lp(M) |p=1 = Hmin(T ).

59

Remark 8.15 Suppose that τ is normalized. We can prove the inequality Hcb,min(T ) 6 0 with

the following reasoning. If ρ is a state, we let σ12def= (Id ⊗ T )(ρ). Using the subadditivity of

the entropy [Pod1, Theorem 8] in the first inequality, we obtain

H[(Id ⊗ T )(ρ)

]− H

[(Id ⊗ τ)(ρ)

]= H(σ12) − H(σ1) 6 H(σ2) 6 0.

Acknowledgement. The author acknowledges support by the grant ANR-18-CE40-0021(project HASCON) of the French National Research Agency ANR. I will thank Luis Rodriguez-Piazza, Carlos Palazuelos, Hanna Podsedkowska and Ami Viselter for very short discussionsand Li Gao for a short but very instructive discussion on von Neumann entropy and the physicalsense of capacities of a quantum channel. We also thank Adrian Gonzalez Perez for an excitingdiscussion on the topic of Section 3.3. I will thank Ion Nechita for a very interesting discussionat the conference “Entangling Non-commutative Functional Analysis and Geometry of BanachSpaces” and Adam Skalski for some questions. Finally, I will thank Christian Le Merdy forsome old very instructive discussions on some results of the Appendices.

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