In [Bra12] a quantum analogue of the HAP was established for these free quantum groups, thus yielding a proof of the MAP for the corresponding reducedC∗-algebras

PROPERTY RD AND HYPERCONTRACTIVITY FOR ORTHOGONAL FREE QUANTUM GROUPS

MICHAEL BRANNAN, ROLAND VERGNIOUX, AND SANG-GYUN YOUN

ABSTRACT. We prove that the twisted property RD introduced in [BVZ15] fails to hold for all non Kac type, non amenable orthogonal free quantum groups. In the Kac case we revisit property RD, proving an analogue of theL_p−L₂ non-commutative Khintchine inequality for free groups from [RX16]. As an application, we give new and improved hypercontractivity and ultracontractivity estimates for the generalized heat semigroups on free orthogonal quantum groups, both in the Kac and non Kac cases.

1. INTRODUCTION

Property RD (Haagerup’s inequality) is a fundamental tool in the study of the reduced C^∗- algebra of discrete groups, allowing one to control the operator norm of convolution operators by means of the much simpler`²-norm (see Section 2.4 for more details). It appeared in the seminal paper [Haa79] where it was used in conjunction with Haagerup’s approximation property (HAP) to establish the Metric Approximation Property (MAP) for reducedC^∗-algebras of free groups.

The definition of Property RD was extended to discrete quantum groups in [Ver07] and was proved there to be satisfied by Kac type (unimodular) orthogonal and unitary free quantum groups.

In [Bra12] a quantum analogue of the HAP was established for these free quantum groups, thus yielding a proof of the MAP for the corresponding reducedC^∗-algebras. Property RD was more- over used for the study of other aspects of discrete quantum group operator algebras, see e.g.

[VV07, Ver12, Bra14, You18]. Interesting connections to Quantum Information Theory, specific to the quantum framework, were also unveiled in [BC18].

The definition of Property RD used in [Ver07] can only be satisfied by Kac type discrete quantum groups. In [BVZ15], the authors give a ”twisted” version of the definition which holds for all (duals of)q-deformations of connected compact semi-simple Lie groups, and give applications to noncommutative geometry.

Hypercontractivity describes the regularization effect, in terms ofL_p-norms, of a given Markov semigroup. It has been studied extensively since the early 70’s, starting with the work of Nelson and Gross [Nel73, Gro72], and has found surprising applications in harmonic analysis, information theory and statistical mechanics. In the case of the Ornstein-Uhlenbeck semigroup on the Clifford algebra with one generator, the two-point inequality of Bonami, rediscovered by Gross [Bon70, Gro75], already has deep applications to (quantum) information theory [BRSdW12, GKK⁺09, KR11, KV15].

In the noncommutative framework, hypercontractivity problems for Orstein-Uhlenbeck-like semi- groups emerged from quantum field theory and optimal times have been obtained in the fermionic

2010Mathematics Subject Classification. Primary 47A30, 43A15; Secondary 20G42, 47D03.

Key words and phrases. Orthogonal free quantum groups, rapid decay property, hypercontractivity.

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case in [Nel73, CL93], using noncommutativeL_p-theory. Moving further away from the commuta- tive situation, hypercontractivity results for free group algebras were obtained in [Bia97, JPP⁺15]

(with respect to different semigroups). Note that the connection between hypercontractivity and Property RD in that case was already noticed by Biane [Bia97].

The study of hypercontractivity for discrete quantum group algebras was initiated in [FHL⁺17], where a natural analogue of the heat semigroup on the reduced C^∗-algebra of orthogonal free quantum groups was studied. In the Kac case, the authors of [FHL⁺17] obtain the ultracontractivity of these semigroups (at all times), as well as hypercontractivity with explicit upper bounds for the optimal time to contractivity.

In the present article we pursue the study of Property RD for non Kac type discrete quantum groups. We prove that non Kac and non amenable orthogonal free quantum groups do not satisfy the property RD introduced in [BVZ15] (Theorem 3.3). Then we state and prove a weaker RD inequality (Proposition 3.4) which holds for all orthogonal free quantum groups, and which was already used without proof in [VV07] in a slightly less precise form.

In the second part of the article we continue the study of ultra- and hypercontractivity for the heat semigroup on free orthogonal quantum groups. We obtain in particular the first known results in the non Kac case, namely ultracontractivity with astrictly positiveoptimal time (Proposition 4.1) and hypercontractivity for large time (Proposition 4.2). In the Kac case we sharpen the upper bound of [FHL⁺17] for the optimal time to hypercontractivity (Theorems 4.5, 4.6 and 4.7), using a non-commutative Khintchine type inequality (Theorem 4.4). We give as well a lower bound for the optimal time to hypercontractivity (Lemma 4.3). Motivated by these results, we end the article with a conjectural formula for the asymptotical behavior of the optimal time to hypercontractivity when the rank of the free orthogonal quantum group tends to infinity.

The article is organized as follows. In Section 2 we recall the necessary preliminaries about compact quantum groups and Property RD on their duals. Section 3 is devoted to the study of Property RD on non Kac type orthogonal free quantum groups. Finally in Section 4 we produce applications to hypercontractivity as described above.

Acknowledgments. M.B. was supported by NSF grant DMS-1700267. R.V. was partially sup- ported by the ANR project ANR-19-CE40-0002. S-G.Y. was supported by the Natural Sciences and Engineering Research Council of Canada and by the National Research Foundation of Ko- rea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01009681). R.V.

thanks Holger Reich and the Algebraic Topology group at the Freie Universit¨at Berlin for their kind hospitality in the academic year 2019–2020.

2. PRELIMINARIES

We assume that the reader is familiar with the basic notation and terminology on compact and discrete quantum groups. For details, we refer the reader to the standard references [Wor98, Tim08, NT13, MVD98]. In this paper we will mainly be concerned with the class of free orthogonal quantum groups and their associated dual discrete quantum groups. We now recall these objects.

2.1. Compact quantum groups. A compact quantum groupG is given by a Woronowicz C^∗- algebraC(G), which is in particular a unital Hopf-C^∗-algebra with co-associative coproduct∆ : C(G) → C(G)⊗C(G). We denote byh the Haar state on C(G), which is the unique state on

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C(G)satisfying

(h⊗id)∆ = (id⊗h)∆ =h(·)1.

The Haar state induces the inner producthf, gi = h(f^∗g)and the normkfk₂ = h(f^∗f)^1/2 forf, g ∈ C(G). By completion we obtain the GNS spaceL₂(G)with canonical cyclic vectorξ₀, and we denoteπ_h :C(G) →B(L₂(G))the associated representation. The image ofπ_h is the reduced Woronowicz C^∗-algebra denoted C_r(G) and the associated von Neumann algebra is L∞(G) = C_r(G)⁰⁰ ⊂B(L₂(G)).

We then defineL₁(G)as the predual ofL∞(G)and consider the natural embeddingL∞(G),→ L₁(G)given byx7→h(·x). Then(L∞(G), L₁(G))is a compatible pair of Banach spaces, which allows one to define the non-commutativeL_p-spacesL_p(G) = (L∞(G), L₁(G))_1/pby the complex interpolation method [Pis03]. When the Haar state is tracial we havekakLp(G) =h(|a|^p)^1/pfor any 1≤p <∞anda∈L_∞(G).

A representation ofGon a Hilbert spaceH_v is an invertible elementv ∈ M(K(H_v)⊗C(G)) such that (id⊗∆)(v) = v₁₂v₁₃, using the leg-numbering notation. Here, v₁₂ = v ⊗1and v₁₃ = σ₂₃(v₁₂)inM(K(H_v)⊗C(G)⊗C(G)), whereσ₂₃is the unique extension of the∗-homomorphism onK(Hv)⊗C(G)⊗C(G)given byT ⊗a⊗b 7→T ⊗b⊗a.

Furthermore,v is called a unitary representation ifv^∗v = IdHv ⊗1C(G) =vv^∗. IfHv is finite- dimensional and equipped with an orthonormal basis(e_i)_i, the associated matrix elements ofvare v_ij = (e^∗_i ⊗id)v(e_j ⊗id). Then we havev = P

e_ie^∗_j ⊗v_ij and∆(v_ij) = P

v_ik ⊗v_kj. For two unitary representationsv ∈M(K(H_v)⊗C(G))andw∈M(K(H_w)⊗C(G)), the tensor product representation isv ^> w=v₁₃w₂₃ ∈M(K(H_v⊗H_w)⊗C(G)).

Furthermore, we say thatvisirreducibleifMor(v, v) :={T ∈B(H_v) :v(T ⊗1) = (T ⊗1)v}= C·id_Hv. We denote byIrr(G)the set of all irreducible unitary representations ofGup to unitary equivalence. For eachα ∈Irr(G)we choose u=u^α ∈αand denote H_α =H_u (which is always finite-dimensional). The coefficients of u^α with respect to some orthonormal basis (e_i)_i ⊂ H_α are denotedu^α_i,j. The multiplicity of an irreducible representationuin another representationv is mult(u⊂v) = dim Mor(u, v).

There is, for each irreducible unitary representationu, a uniquely defined positive elementQ_u ∈ B(H_u) such thatd_u := Tr(Q_u) = Tr(Q⁻¹_u ) and such that the following orthogonality relations hold

h(u^∗_iju_kl) = d⁻¹_u δ_jl(e_k |Q⁻¹_u e_i),

h(uklu^∗_ij) = d⁻¹_u δik(ej |Quel). (2.1) The numberd_u is called thequantum dimensionofu, as opposed to the classical dimensionn_u = dimH_u. The compact quantum groupGis said to be ofKac typeifQ_α= id_Hα for allα∈Irr(G).

This is equivalent to the Haar statehbeing tracial.

The coefficients u^α_i,j of irreducible unitary representations span a dense subalgebra O(G) ⊂ C(G)which is a Hopf algebra with respect to the restriction of the coproduct ∆. We recall that h is faithful on O(G) and we shall identify O(G) with its image in Cr(G) through the GNS representation π_h — in particular we identify a representation v and its image (id⊗π_h)(v) ∈ B(H_v)⊗C_r(G). Note also thatO(G)is dense inL_p(G)for any1≤p < ∞.

A compact quantum groupGis said to be a compact matrix quantum group if there exists a finite generating subset{α₁,· · · , α_n}ofIrr(G)in the sense that any irreducible unitary representation u^αappears as an irreducible component of a tensor product representationu^{αm1 >} u^{αm2 >} · · · ^>u^αmk

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for somek∈Nand1≤m₁, m₂,· · · , m_k ≤n. In this case, for anyα, the minimal numberk ∈N⁰ required to generate u^α as a subrepresentation as above is called the length of α, and denoted

|α| = k. The length at the trivial representation is 0. We say that a non-zero elementf ∈ C(G) or C_r(G) has length k if it can be written as a linear combination of coefficients u^α_i,j with irre- ducible representationsαof lengthk. We denotep_k ∈ B(L₂(G))the orthogonal projection onto the subspace ofL₂(G)spanned by elements of lengthk.

2.2. Dual algebras. Associated to each compact quantum groupG is its dual discrete quantum groupGb. For us the main object of interest will be the algebra

`∞(Gb) ={a ∈Q

α∈Irr(G)B(Hα) : (kaαk)α bounded}

and the subalgebrasc₀₀(Gb),c₀(Gb)of sequences with finite support, resp. converging to0. For each α ∈ Irr(G) we denotep_α the corresponding minimal central projection in any of these algebras.

We use the same notation p_α for the orthogonal projection onto the subspace of L₂(G)spanned by the GNS images of the coefficientsu^α_i,j — indeed there is a natural representation ofc₀(bG)on L₂(G)which realizes this identification.

The algebras c0(Gb) andC(G)are related through the “multiplicative unitary” V = L

αu^α ∈ M(c₀(Gb)⊗C(G)). We endowc₀(bG)and`∞(bG)with the coproduct∆ˆ such that( ˆ∆⊗id)(V) = V₁₃V₂₃. By definition this coproduct is related to the tensor product construction for representa- tions, more precisely we have, for allα, β, γ ∈ Irr(G), a ∈ B(H_γ)andT ∈ Mor(γ, α ^> β), the following identity inB(H_γ, H_α⊗H_β):

(p_α⊗p_β) ˆ∆(a)T =T a.

There is a distinguished weightˆhon`_∞(Gb), called theleft Haar weight, given by ˆh(a) = X

α∈Irr(G)

d_αTr(Q_αa_α) (a= (a_α)_α ∈c₀₀(Gb)).

We denote againkak₂ = ˆh(a^∗a)^1/2the norm onc₀₀( ˆG)associated with this weight. By restriction and tensor product one obtains as well norms, still denoted k · k₂, on B(H_α) andB(H_β ⊗H_γ), associated to the inner productsha₁, a₂i =d_αTr(Q_αa^∗₁a₂)for alla₁, a₂ ∈ B(H_α)andhx₁, x₂i = d_βd_γTr((Q_β ⊗Q_γ)x^∗₁x₂) for all x₁, x₂ ∈ B(H_β ⊗H_γ). Note that the collection of matrices Q_α defines an algebraic (in general unbounded) multiplierQ= (Q_α)_αofc₀₀( ˆG), themodular element.

The analogue of the classical Fourier transform is the linear mapF :c₀₀(Gb)→C(G)given by F(a) = (ˆh⊗id)(V(a⊗1)). Explicitly, we have

F(a) = X

α∈Irr(G) nα

X

i,j=1

d_α(a_αQ_α)_j,iu^α_i,j ∈C_r(G).

The Haar statehonGand the left Haar weightˆhonGb are related through the Plancherel Theorem, which asserts that for anya = (aα)α∈Irr(G) ∈c00(Gb), we haveˆh(a^∗a) = h(F(a)^∗F(a)).

Let us note the following algebraic properties of the Fourier transform. Recall that for f ∈ O(G), ϕ ∈ O(G)^∗ we denotef ∗ϕ = (ϕ⊗id)∆(f)andϕ∗f = (id⊗ϕ)∆(f). Then we have, fora ∈c₀₀(bG),ϕ∈ O(G)^∗:

ϕ∗ F(a) = F(ba) and F(a)∗ϕ=F(ab^Q)

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whereb = (id⊗ϕ)(V)is an algebraic multiplier ofc₀₀(Gb)andb^Q = QbQ⁻¹. On the other hand fora,b∈c₀₀(Gb)we haveF(a)F(b) =F(a ? b)wherea ? bis the unique element ofc₀₀(Gb)such that(ˆh⊗ˆh)(∆(c)(ab ⊗b)) = ˆh(c(a ? b))for allc∈c₀₀(Gb). The mapa⊗b 7→a ? bdefined above is referred to as theconvolution productonc₀₀(Gb).

We say thatGb is finitely generated whenGis a compact matrix quantum group. Having fixed a generating subset inIrr(G), we putp_n=P

|α|=np_α ∈c₀₀( ˆG). This is compatible with the notation p_n ∈ B(L₂(G))introduced previously, in the sense that we haveF(p_na)ξ₀ = p_nF(a)ξ₀ for any n∈N0 anda ∈c₀₀( ˆG).

2.3. The free orthogonal quantum groups. We now come to the main objects of study in this paper. LetN ∈N, N ≥2andF ∈ GL_N(C)such thatF F = ±1. Thefree orthogonal quantum groupis the compact quantum groupO⁺_F = (C(O_F⁺),∆), where

(1) C(O_F⁺) is the universal unital C^∗-algebra generated byN² elements u_i,j, 1 ≤ i, j ≤ N, satisfying the relations making u unitary and u = (F ⊗ 1)u^c(F⁻¹ ⊗ 1), where u = (u_i,j)_1≤i,j≤N ∈M_N(C)⊗C(O⁺_F)andu^c= u^∗_i,j

1≤i,j≤N.

(2) ∆ : C(O_F⁺) →C(O⁺_F)⊗C(O_F⁺)is the unital∗-homomorphism determined by∆(u_i,j) = PN

k=1u_i,k ⊗u_k,j.

The compact quantum groupO⁺_F is a compact matrix quantum group and we choose thefunda- mental representationu= (u_i,j)_i,j ∈B(C^N)⊗C(O⁺_F), coming from the canonical generators of C(O_F⁺), as the (unique) generating representation. Then it is known from [Ban96] that for each k ∈N0there is a unique irreducible representation (up to equivalence) of lengthk, which is equiv- alent to its conjugate. We denote this classk, yielding an identification ofIrr(O_F⁺)with N0. We have in particularu⁰ = 1_C(G) (thetrivial representation), andu¹ = u = (u_i,j)∈ B(H₁)⊗C(G) withH₁ =C^N.

One can check thatQ1 = F^trF¯, so that d1 = Tr(F^∗F). There exists a uniqueq ∈ (0,1]such thatd1 = q+q⁻¹ and we denote also Nq = d1 = q+q⁻¹. On the other hand one can see that kQ_kk=kQ₁k^k =kFk^2kfor allk∈N, and thatO⁺_F is of Kac type iffF is unitary. This is typically the case ofF =I_N and we denote in this caseO⁺_N :=O⁺_I

N.

It is moreover known thatu^{m >} uⁿis unitarily equivalent to u^|m−n|⊕u^|m−n+2|⊕ · · · ⊕u^m+n. We denote byPl = P_l^m,n the orthogonal projection from Hm ⊗Hn ontoHl for any one ofl =

|m−n|,|m−n|+ 2,· · · , m+n. We have in particularn0 = 1, n1 = N, n1nk+1 = nk+2 +nk

for allk ∈ N⁰ andd₀ = 1,d₁ = N_q := Tr(F^∗F),d₁d_k+1 = d_k+2+d_k for allk ∈ N⁰. Finally, it was also shown by Banica [Ban96] that thefundamental characterχ₁ =PN

i=1u_iiis a semicircular element (on[−2,2]) with respect to the Haar state.

2.4. Property RD and its generalizations. In the case whenGb is a classical discrete groupΓ, the Property of Rapid Decay amounts to controlling the norm ofC_r(G) = C_r^∗(Γ)from above by the 2-norm. More precisely a discrete groupΓ has Property RD if there exists a polynomialP such that

kxkC^∗_r(Γ)≤P(k)kxk2 (2.2)

for all k ∈ N0 and all x ∈ C_r^∗(Γ)supported on elements of lengthk inΓ, with respect to some fixed length (for instance a word length ifΓis finitely generated). Note that the reverse inequality kxk2 ≤ kxk_Cr^∗_(Γ)is always true.

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A quantum generalization of Property RD was introduced in [Ver07] by means of the same inequality (2.2), with appropriate notions of length and support as introduced above. It was shown in the same article that Property RD holds for the dual ofO⁺_N but fails for the dual of any compact quantum groupG which is not of Kac type. Later, a modification of the quantum definition was proposed in [BVZ15] so as to accommodate non-Kac examples such asSU_q(2), and more generally quantum groupsGwith (classical) polynomial growth. This modification is obtained by replacing the2-norm on the right-hand side of (2.2) by a still “easily computable” twisted2-norm.

In this setting, “easily computable” means a norm of the form kfk_ϕ = kϕ ? fk₂ orkf ? ϕk₂ forf ∈ O(G), withϕ∈ O(G)^∗ fixed. Using the fact that the Fourier transform is isometric, this can also be writtenkF(a)k_ϕ = kDak₂ orkaD^Qk₂ fora ∈ c₀₀( ˆG), whereD = (id⊗ϕ)(V), and these norms can indeed be computed by multiplying matrices and summing their traces. In this picture the twisted Property RD takes the form kF(a)k_Cr(G) ≤ P(k)kDak₂ or kF(a)k_Cr(G) ≤ P(k)kaD⁰k₂ifF(a)is of lengthk, for some fixed algebraic multiplierDorD⁰ofc₀₀(Gb). Observe that by polar decomposition one can assume D > 0 (resp. D⁰√

Q > 0) without changing the associated twisted norm.

Of course one could always achieve such inequalities by taking a central multiplierD= (b_αI_Hα)_α with weightsb_αgrowing sufficiently rapidly (see for example [VV07] and the discussion at the be- ginning of Section 3.1). However, for some applications (e.g., to the metric approximation property [Bra12], and to non-commutative geometry [BVZ15]), it is desirable to use “natural” or “optimal”

elementsD,D⁰.

We note that the authors of [BVZ15] choose the twisted2-norm in such a way that√

nαu^α_i,j 1≤i,j≤nα

α∈Irr(G)

forms an orthonormal basis, as it is in the case of Kac type compact quantum groups. An easy in- spection with our conventions shows that the only twisted norm with this property iskF(a)k_ϕ :=

ka√

Ck₂, where

C =d_α nα

Q_α

α (2.3)

is the canonical element used in [BVZ15] to define their twisted2-norms. In the following defi- nition we fix a multiplierD = (D_α)α∈Irr(G) of c₀₀(Gb), we consider the associated twisted norms kak2,D :=kaDk2 fora∈c00(Gb), and we put

kfk_2,D :=kF⁻¹(f)k_2,D =kF⁻¹(f)Dk₂

forf ∈ O(G). Observe thatDis uniquely determined byk · k_2,Dif we assumeD√

Q≥0.

Definition 2.1. LetGbe a compact matrix quantum group with a fixed family of generating irre- ducible representations andD a multiplier of c₀₀(bG). We say thatGb has PropertyRD_D if there exists a polynomialP ∈R⁺[X]such that for allk ∈N⁰ andf ∈ O(G)of lengthk, we have

kfk_Cr(G) ≤P(k)kfk_2,D.

The property above can also be written kF(a)k_Cr(G) ≤ P(k)kak_2,D for all k ∈ N0 and a ∈ p_kc₀₀(Gb). Explicitly, Property RD_D asks that

k X

|α|=k nα

X

i,j=1

d_α(a_αQ_α)_j,iu^α_i,jk²_Cr(G)≤P(k)² X

|α|=k

d_αTr(D_αQ_αD_α^∗a^∗_αa_α). (2.4)

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The Property RD considered in [BVZ15] corresponds to the case D = √

C, which satisfies D√

Q≥0sinceCcommutes withQ. IfGis of Kac type, the PropertyRD^√_C coincides with the propertyRDin [Ver07]. In particular, if Gb is a discrete groupΓ, then PropertyRD^√_C is exactly same with the property RD ofΓ.

We now restate [Ver07, Lemma 4.6] in a slightly more general form. Note that in the case G = O_F⁺ equipped with the canonical generating representation, there is only one irreducible representation α = k ∈ Irr(G) for each given length k, and the inclusions u^l ⊂ u^k ⊗uⁿ are multiplicity-free.

Lemma 2.2. Let G be a compact matrix quantum group with a fixed family of generating irre- ducible representations. Fork,n∈N0 andγ ∈Irr(G)we denote

ν_k,n^γ = X

|α|=k,|β|=n

dαdβ

d_γ mult(u^γ ⊂u^{α >}u^β). (2.5) Then the discrete quantum groupGbhas PropertyRD_Dwith respect to a multiplierDiffthere exists a polynomialP such that we have, for anyk,l,n ∈N⁰and for everya∈p_kc₀(bG),b ∈p_nc₀(Gb):

X

|γ|=l

ν_k,n^γ k∆(pˆ γ)(a⊗b) ˆ∆(pγ)k²₂ ≤P(k)²kaD⊗bk²₂. (2.6) Proof. The proof is a straightforward extension of the ideas in the proof of [Ver07, Lemma 4.6]

using our notation. Let us recall the main ideas for the convenience of the reader.

First of all, Property RD_D is equivalent to the fact thatkp_lf p_nk_Cr(G) ≤ P(k)kfk_2,D for all k, l,n ∈ N0 andf ∈ O(G)of lengthk, see [Ver07, Proposition 3.5] and [BVZ15, Proposition 3.4].

Using the Fourier transform, this means that we require

kp_lF(a)F(b)ξ₀k₂ ≤P(k)kak_2,Dkbk₂ =P(k)kaD⊗bk₂, (2.7) for allaof lengthk andbof lengthn. Moreover we havekp_lF(a)F(b)ξ₀k₂ =kp_lF(a ? b)ξ₀k₂ = kF(pl(a ? b))k2 =kpl(a ? b)k2 — indeed by definition ofpl(inc0(Gb)andB(L2(G))) and ofV we have(1⊗pl)V(1⊗ξ0) = (pl⊗id)V(1⊗ξ0). Then we can decompose into orthogonal components:

kp_l(a ? b)k²₂ =P

|γ|=lkp_γ(a ? b)k²₂.

Then by definition of the convolution product we can write, for anyc∈c₀(Gb):

ˆh(c^∗p_γ(a ? b)) = (ˆh⊗ˆh)( ˆ∆(c)^∗∆(pˆ _γ)(a⊗b))

= (ˆh⊗ˆh)((p_k⊗p_n) ˆ∆(c)^∗∆(pˆ _γ)(a⊗b) ˆ∆(p_γ)).

Note that pγ is central in c0(Gb)and that ∆(pˆ γ) is (ˆh⊗ˆh)-central. To obtain the expression of kpγ(a ? b)k²₂ which appears in the left-hand side of (2.6), it remains to take the supremum over c∈ pγc0(Gb), with kck2 ≤ 1. We show below that we have in factk(pk⊗pn) ˆ∆(c)k²₂ =ν_k,n^γ kck²₂, which yields the correct expression kp_γ(a ? b)k₂ = (ν_k,n^γ )^1/2k∆(pˆ _γ)(a⊗b) ˆ∆(p_γ)k₂ so that (2.6) results from (2.7).

Indeed we have∆(Q) =ˆ Q⊗Qin the multiplier algebra ofc₀₀(Gb)⊗c₀₀(Gb), and on the matrix algebra p_γc₀(Gb) = B(H_γ) the ∗-homomorphism(p_α ⊗p_β) ˆ∆is an amplification with the same multiplicity as the inclusionu^γ ⊂u^{α >} u^β. Thus we have

Tr

(Q⊗Q)(p_α⊗p_β) ˆ∆(d)

=mult(u^γ ⊆u^{α >}u^β) Tr(Qd) (2.8)

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for anyd∈B(H_γ). As a result we can write (ˆh⊗ˆh)

(p_k⊗p_n) ˆ∆(p_γd)

= X

|α|=k,|β|=n

d_αd_β(Tr⊗Tr)[(p_α⊗p_β)(Q⊗Q) ˆ∆(p_γd)]

= X

|α|=k,|β|=n

d_αd_βmult(u^γ ⊂u^{α >} u^β) Tr(Qp_γd) =ν_k,n^γ ˆh(p_γd).

Takingd=c^∗cwe obtaink(p_k⊗p_n) ˆ∆(c)k²₂ =ν_k,n^γ kck²₂as claimed.

3. ON PROPERTY RDD FOROc⁺_F

In this section we turn our attention to the duals of the free orthogonal quantum groups O_F⁺, establishing some necessary conditions for Property RD_D to hold for a given multiplierD. In this case Property RD_D with respect to a multiplier D and a polynomial P is characterized by the following multiplicity-free version of (2.6):

k∆(pˆ _l)(a⊗b) ˆ∆(p_l)k₂ ≤ r d_l

d_kd_nP(k)kaD_k⊗bk₂, (3.1) for allk,l,nsuch thatu^l⊂u^k⊗uⁿ,a ∈B(Hk)andb∈B(Hn).

Here the 2-norms are the ones coming from the weightˆh, but one can use as well the twisted Hilbert-Schmidt norms, e.g.kak²_HS = Tr(Q_ka^∗a)fora∈B(H_k), since these two norms only differ by a scalar factor √

d_k. Moreover, if we fix an isometric intertwiner (unique up to a phase) v = v_l^k,n ∈Mor(u^l, u^k⊗uⁿ)we havek∆(pˆ _l)(a⊗b) ˆ∆(p_l)k_HS =kvv^∗(a⊗b)vv^∗k_HS=kv^∗(a⊗b)vk_HS

— notice that the last norm is the twisted Hilbert-Schmidt norm onB(H_l).

We can moreover give an explicit form to the intertwinersv_l^k,n as follows. For each n ∈ N⁰ the tensor power representationu¹ > · · · >u¹ contains a unique copy ofuⁿ, we choose forH_nthe corresponding subspace ofH₁^⊗nand we denoteP_n =p^⊗n₁ ∆ˆⁿ⁻¹(p_n)∈B(H₁^⊗n)the corresponding orthogonal projection. We further fix an intertwiner (unique up to a phase)t_n ∈ Mor(1, u^{n >} uⁿ) such that kt_nk = √

d_n and we consider the intertwiners A^k,n_l = (P_k ⊗P_n)(id⊗t_r ⊗id)P_l ∈ Mor(u^l, u^k⊗uⁿ), wherer = (k+n−l)/2. One can then takev^k,n_l =kA^k,n_l k⁻¹A^k,n_l .

The (operator) norm ofA^k,n_l can be explicitly computed, see for example [Ver07, Lemma 4.8]

or [BC18, Equation (6) and Proposition 3.1]. This norm happens to be controlled from below and above, up to factors depending only the parameter0 < q < 1given byq +q⁻¹ = Tr(F^∗F), as follows:

1

d_r ≤ kA^k,n_l k⁻² = 1 d_r

r

Y

s=1

(1−q^2+2s)(1−q^2l−2r+2s)(1−q^2m−2r+2s) (1−q^2k+2+2s)(1−q^2s)²

≤ 1 [r+ 1]_q

Y^r

s=1

1 1−q^2s

3

≤ 1 d_r

Y^∞

s=1

1 1−q^2s

3

,

wherel =k+n−2r. If we put

1< C(q) = 1 (1−q²)

Y^∞

s=1

1 1−q^2s

3

, (3.2)

8

and use the inequality

(1−q²)³ ≤ d_kd_n

d_ld²_r ≤(1−q²)⁻² which follows from the dimension formuladn = ^qn+1_q−q^−q−1⁻ⁿ⁻¹, we get

(1−q²)^3/2 d_l d_kd_n

1/2

≤ kA^l,m_k k⁻² ≤C(q) d_l d_kd_n

1/2

. (3.3)

Inequality (3.3) shows thatkA^k,n_l k⁻² compensates exactly for the analogous factor of the right- hand side of theRDinequality (3.1) which can thus be rewritten in the equivalent formulation

k(A^k,n_l )^∗(a⊗b)A^k,n_l k_HS≤P(k)kaD_k⊗bk_HS. (3.4) In (3.4), it is important to use the twisted Hilbert-Schmidt norms since the matrix spaces are no longer the same on both sides.

Remark. The universal constantC(q)defined in (3.2) will make several appearances in the remain- der of the paper.

Note that we have u^l ⊂ u^k >uⁿ iff l ∈ {|n−k|,|n −k + 2|, . . . , n +k}. One can obtain necessary conditions for PropertyRD_D by fixing the value ofl. More specifically we say that the dual ofO⁺_F satisfies PropertyRD⁰_D (resp. RD^max_D ) for some polynomialP if the above inequality is satisfied for allk=n∈N0 andl = 0(resp. for allk,n ∈N0 and forl=k+n).

In the case ofRD_D⁰ we have simplyA^k,n_l =A^n,n₀ =t_n: C→H_n⊗H_nso that PropertyRD⁰_D, with respect to the polynomialP, is equivalent to the fact that

|t^∗_n(a⊗b)t_n| ≤P(n)kaD_n⊗bk_HS. (3.5) for alln ∈N0 anda,b ∈B(H_n). Note thatt_ncan be written uniquely ast_n(1) = P

ie_i⊗j_n(e_i), where the anti-linear mapj_n : H_n → H_ndoes not depend on the chosen orthonormal basis (e_i)_i ofH_n, and recall that we have Q_n = j_n^∗j_n andj_n² = ±id. Then one can compute (recalling that (ζ|jξ) = (ξ|j^∗ζ)for an anti-linear mapj):

t^∗_n(a⊗b)t_n = Tr(j_n^∗b^∗j_na) (3.6) Summing everything up, one can reformulate PropertyRD⁰_D as follows:

Definition 3.1. We say thatOc⁺_F hasRD⁰_D if there exists a polynomialP such that

|t^∗_n(a⊗b)t_n|= Tr(j_n^∗b^∗j_na)≤P(n)kaD_n⊗bk_HS (3.7) for alla, b∈B(H_n)andn ∈N0.

It turns out that PropertyRD⁰_D for the dual ofO⁺_F can be explicitly characterized in terms of the matricesD_n(andQ_n), as follows.

Proposition 3.2. The discrete quantum groupOc⁺_F hasRD⁰_D with respect to a polynomialP if and only if

kQ^−1/2_n D⁻¹_n QnkkQ^1/2_n k ≤P(n)f or all n ∈N. (3.8)

9

Proof. Note that (3.7) can be written as

|Tr(b^∗a)|² ≤P(n)²kaD_n⊗j_n^∗bj_nk²_HS=P(n)²Tr(D_n^∗Q_nD_na^∗a) Tr(Q_nj_n^∗b^∗j_nj_n^∗bj_n).

We have moreoverD^∗_nQ_nD_n = ˜D²_n, where D˜_n =D_n√

Q_n is positive, andTr(Q_nj_n^∗b^∗j_nj_n^∗bj_n) = Tr(j_nj_n^∗j_nj_n^∗b^∗j_nj_n^∗b) = Tr(Q⁻¹_n b^∗Q⁻¹_n bQ⁻¹_n ). Now we note that, by the Cauchy-Schwarz inequal- ity (for the untwisted Hilbert-Schmidt scalar product), the maximum of |Tr(b^∗a)|²/Tr( ˜D_n²a^∗a) equalsTr( ˜D⁻²_n b^∗b), attained ata=bD˜⁻²_n , so thatRD⁰_D is equivalent to

Tr( ˜D_n⁻²b^∗b)≤P(n)²Tr(Q⁻¹_n b^∗Q⁻¹_n bQ⁻¹_n ) ReplacingbwithQ

1

n2bQ_n, the above can be written as

Tr(QnD˜_n⁻²Qnb^∗Qnb)≤P(n)²Tr(b^∗b).

We note that, for positive matricesM, N ∈B(H_n)₊, maxb6=0

Tr(M b^∗N b)

Tr(b^∗b) = max

b6=0

kN¹²bM¹²k_HS kbk_HS

!2

equals kMkkNk, attained at b = ξη^∗ ∈ B(H_n), where ξ, η are unit vectors chosen to satisfy kN¹²ξk=kN¹²kandkM¹²ηk=kM¹²k. Therefore,RD_D⁰ is equivalent to

kQ_nD˜_n⁻²Q_nkkQ_nk=kD˜⁻¹_n Q_nk²kQ^1/2_n k² ≤P(n)² for alln ∈N.

which gives us the desired conclusion.

Remark. The same techniques apply if one tries to twist the 2-norm from the other side, i.e. if one considers inequalities of the form|t^∗_n(a⊗b)tn| ≤P(n)kDna⊗bkHS. Then one arrives at the conditionkD⁻¹_n Q^1/2n kkQ^1/2n k ≤P(n), which is equivalent to the condition of Proposition 3.2 ifD andQcommute.

Recall that [BVZ15] take D_k² = C_k = ^d_n^k

kQ_k to verify RD_D for SU_q(2) (more generally, the Drinfeld-Jimboq-deformationsG_q). It seems reasonable to consider a continuous family of vari- ants of this multiplier by takingD_k = (^d_n^k

k)^|s|/2Q^s/2_k withs∈ R. Recalling thatkQ_kk=kQ⁻¹_k k= kQ₁k^k, we see that in that case Proposition 3.2 reads

n_k d_k

|s|

kQ₁k^(|1−s|+1)k ≤P(k)² for allk ≥0.

However, the following theorem shows that this inequality is not satisfied for any non-KacO⁺_F as soon asN ≥3.

Theorem 3.3. Let N ≥ 3, s ∈ R and consider the multiplier D(s) = (Dk)k∈N0, with Dk = (d_k/n_k)^|s|/2Q^s/2_k . ThenOc_F⁺has PropertyRD⁰_D(s)if and only ifO_F⁺is of Kac type. In particular, all non-KacO⁺_F do not have propertyRD_D(s).

Proof. We only need to consider the case whereO⁺_F is not of Kac type. I.e., we assumeQ₁ 6=I. Since

n_k dk

|s|

kQ₁k^(|1−s|+1)k ≥

n_k

dkkQ₁k^k|s|

for allk ∈ N, it suffices to show that ⁿ_d^k

kkQ₁k^k has exponential growth, i.e.

lim inf

k→∞

n_kkQ₁k^k d_k

¹_k

>1.

10

First of all, we havelimk→∞d

1 k

k = f(d₁)andlimk→∞n

1 k

k = f(n₁)wheref(t) = ^t+

√ t²−4 2 . Let us denote byλ₁ ≤ · · · ≤ λ_N the eigenvalues ofQ₁. Then, since the spectrum ofQ₁ is symmetric under inversion, we haveλ₁ <1andλ₂ ≤1.

Then, in the expansion of(λ₁+· · ·+λ_N)² =PN

i,j=1λ_iλ_j, we haveλ²₁, λ₁λ₂, λ₂λ₁ <1,λ²₂ ≤1 and the other terms are smaller thanλ²_N. From this observation we obtain

d²₁ <4 + (N²−4)λ²_N, which together with the obvious estimated₁ < N λ_N yields

f(d₁) = d₁+p d²₁−4

2 < N λN +p

(N²−4)λ²_N

2 =f(n₁)kQ₁k.

Hence, we havelim infk→∞

nkkQ_kk dk

_k¹

= ^f(n_f(d^1)kQ1k

1) >1.

Remark. On the other hand, one might try to consider the ”opposite” case of RD_D^max, which is satisfied iff we havekPk+n(a⊗b)Pk+nkHS≤P(k)kaDk⊗bkHSfor allk, n∈N⁰anda∈B(Hk), b ∈B(H_n). In fact, whenDcommutes withQit turns out thatRD^max_D is a consequence ofRD⁰_D, thank to Proposition 3.2.

Indeed we always have kP_k+n(a ⊗b)P_k+nk_HS ≤ ka ⊗bk_HS (using the fact that P_k+n com- mutes with Q_k ⊗ Q_n), so that kak_HS ≤ P(k)kaD_kk_HS implies RD_C^max. Performing the same analysis as in the proof of Proposition 3.2 we see that this stronger condition is equivalent to kD⁻¹_k k ≤ P(k)for all k. On the other hand whenD commutes with Qwe can writekD⁻¹_k k = kQ^−1/2_k D_k⁻¹Q_kQ^−1/2_k k ≤ kQ^−1/2_k D_k⁻¹Q_kkkQ^−1/2_k k, which makes the connection with the charac- terization ofRD⁰_D given at Proposition 3.2 sincekQ^−1/2_k k=kQ^1/2_k k.

Remark. The analysis of the above two subcases of PropertyRD_D leads us to ask whether Prop- erty RD_D is equivalent to RD_D⁰ for Oc⁺_F? I.e., Is Property RD_D equivalent to the inequalities kQ^−1/2_k D_k⁻¹Q_kkkQ^1/2_k k ≤P(k), at least whenDandQcommute?

3.1. A Weaker Variant of Property RD. Despite the failure ofRD^√_C for non-amenable, non- Kac type orthogonal free quantum groups, one can prove a weaker RD inequality (corresponding to a larger multiplierD) which holds for all orthogonal free quantum groups, and also for all discrete quantum groups with polynomial growth. This inequality was already stated (without proof) and used in [VV07], see Remark 7.6 therein. We provide below a slightly more precise statement and a proof. In the following section, we will see how this weakened property RD is applicable to find almostsharp optimal time estimates for ultracontractivity of heat semigroups onO_F⁺.

Proposition 3.4. LetF ∈ GLN(C)be such thatFF¯ =±IN, N ≥ 2. Then for any k,l,n ∈ N⁰ anda ∈B(Hk)⊂`∞( ˆO⁺_F)we have

kp_lF(a)p_nk ≤C(q)kFk^2kkak₂ and kF(a)k ≤C(q)(k+ 1)kFk^2kkak₂, whereC(q)>1is the constant defined by(3.2)for0< q <1such thatTr(F^∗F) = q+q⁻¹. Proof. We follow quite closely the proof of [Ver07, Theorem 4.9], taking into account the twist- ing of Hilbert-Schmidt norms. Starting again from (3.1) and taking into account (3.3) as in the beginning of Section 3 the first inequality will follow if we prove

k(A^k,n_l )^∗(a⊗b)A^k,n_l kHS≤ kFk^2kka⊗bkHS (3.9)

11

for any k, l, n ∈ N⁰ such that u^l ⊂ u^k > uⁿ, a ∈ B(H_k) and b ∈ B(H_n). Since P_l is an orthogonal projection, the left-hand side admitsk(id⊗t^∗_r⊗id)(a⊗b)(id⊗t_r⊗id)k_HSas an evident upper bound, where r = (k+n −l)/2 and we are using the twisted Hilbert-Schmidt norm on B(Hk−r ⊗Hn−r). (Note that the projection P_l commutes with the matrixQ_k⊗Q_n defining the twisting of the Hilbert-Schmidt norm.)

We decomposea =P

ia_i⊗E_iandb=P

ij_r^∗E_ij_r⊗b_i, wherea_i ∈B(Hk−r),b_i ∈B(Hn−r)and (E_i)_iis the basis of matrix units inB(H_r)corresponding to an orthonormal basis of eigenvectors of Q_rinH_r. With this choice we have in particular that(E_i)_iand(j_r^∗E_ij_r)_iare orthogonal bases with respect to the twisted Hilbert-Schmidt scalar product onB(H_r), and one can moreover compute, ifE_i = e_pe^∗_q ande_p, e_q are eigenvectors ofQ_r with respect to eigenvalues λ_p, λ_q: kE_ik²_HS = λ_q, kj_r^∗E_ij_rk²_HS =λ⁻²_q λ⁻¹_p . In particular we note that

kE_ik⁻²_HS =λ_qλ_pkj_r^∗E_ij_rk²_HS≤ kQ_rk²kj_r^∗E_ij_rk²_HS. (3.10) According to (3.6) we can then write

k(id⊗t^∗_r⊗id)(a⊗b)(id⊗t_r⊗id)k_HS=kP

i,jTr(E_j^∗E_i)(a_i⊗b_j)k_HS=kP

ia_i⊗b_ik_HS. Now we apply the triangle inequality and Cauchy-Schwartz inequality:

k(A^k,n_l )^∗(a⊗b)A^k,n_l k²_HS ≤(P

ika_i⊗b_ik_HS)² = (P

ika_ik_HSkb_ik_HS)²

≤P

ika_ik²_HSkE_ik²_HSP

ikb_ik²_HSkE_ik⁻²_HS =kak²_HSP

ikb_ik²_HSkE_ik⁻²_HS. Finally we have P

ikbik²_HSkEik⁻²_HS ≤ kQrk²P

ikbik²_HSkj_r^∗Eijrk²_HS = kQrk²kbk²_HS by (3.10).

SincekQ_rk=kQ₁k^r=kFk^2r ≤ kFk^2kwe have proved (3.9).

The second inequality in the statement follows from the first one by a standard argument, see [Ver07, Proposition 3.5], using the fact that for anynthe tensor productu^{k >} uⁿhas at mostk+ 1

irreducible subobjects.

Remark. The Property above can be interpreted as PropertyRD_D with respect to thecentralmul- tiplier D = P

k∈N0kFk^2kp_k = P

k∈N0kQ_kkp_k and the (constant) polynomialP = C(q). Note that the element C in [BVZ15] satisfies√

C ≤ D and thanks to [BVZ15, Proposition 4.2] this implies that PropertyRD_D, for this elementD, is also satisfied by all discrete quantum groups of polynomial growth, and still reduces to the usual PropertyRDfor (classical) discrete groups.

4. APPLICATIONS: ULTRACONTRACTIVITY AND HYPERCONTRACTIVITY OF THE HEAT

SEMIGROUP ONO_F⁺

In this section of the paper, we are interested in studying hypercontractivity and ultracontractiv- ity properties of the heat semigroup(T_t)_t>0 on the free orthogonal quantum groupsO_F⁺. This heat semigroup was introduced and studied in [CFK14, FHL⁺17] in the Kac-type setting (i.e.,F =I_N), but a standard argument using results from [DCFY14, Fre13] on monoidal equivalences and trans- ference properties of central multipliers allows one to define an appropriate heat semigroup on all free orthogonal quantum groupsO⁺_F’s. The details of this are spelled out, for example, in [Cas18, Section 6.1].

LetM be a von Neumann algebra equipped with a fixed faithful normal stateϕ. In the following, aϕ-Markov semigroup onM will mean aσ-weakly continuous semigroup(T_t)t≥0of normal unital completely positiveϕ-preserving maps T_t : M → M. With a slight abuse of notation, we will identifyM ⊂ L₂(M)as a dense subspace (via the GNS map associated toϕ) also denote by T_t : L2(M) → L2(M)the canonical extension ofTtto a contraction on the GNS spaceL2(M). The

12