$L_{p}$-improving convolution operators on finite quantum groups

Introduction

which provides a more general method to construct $L_{p}$-improving measures on groups.

In particular, the result characterizes the Fourier-Schur multipliers on finite groups which have an $L_{p}$-improving property. Let $\Gamma$ be a finite group and $\varphi$ be a positive definite function on $\Gamma$. Let $M_{\varphi}$ be the associated Fourier-Schur multiplier operator determined by $M_{\varphi}(\lambda(\gamma))=\varphi(\gamma)\lambda(\gamma)$ for all $\gamma\in\Gamma$. Then

if and only if $|\varphi(\gamma)|<1$ for any $\gamma\in\Gamma\setminus\{e\}$.

We should emphasize that our argument relies essentially on new and interesting properties on the unital trace preserving operators on noncommutative $L_{p}$-spaces, based on the recent work of Ricard and Xu [RX16]. In fact, the following fact proved in Theorem 1.6 plays a key role in our argument. For a finite dimensional C*-algebra $A$ equipped with a faithful tracial state $\tau$, and $T:A\to A$ a unital trace preserving map, the $L_{p}$-improving property

holds if and only if we have the following “spectral gap”:

We provide two proofs of this result, where one is based on very elementary arguments with an additional assumption of $2$-positivity and another, which is rather short, on [RX16]. In Theorem 1.9 we also show that the $L_{p}$-improving property (0.1) remains stable under the free products. This method permits us to give $L_{p}$-improving convolution operators for infinite quantum groups.

In this paper we also include some simple properties of non-degenerate states on compact quantum groups with applications. We prove in Lemma 3.3 that the convolution Cesàro limit of a non-degenerate state is the Haar state, which not only contributes to the proof of our main result, but also yields a generalization of [BFS12, Theorem 2.2] concerning the computation of idempotent states associated to Hopf images.

We end this introduction with a brief description of the organization of the paper. Section 1 deals with the characterization of unital trace preserving $L_{p}$-improving operators on finite dimensional C*-algebras and their free products. In Section 2 we present some preliminaries on compact quantum groups and the related Fourier analysis. Here we give a short and explicit calculation of Fourier series for compact quantum groups, parallel to the case of classical compact groups, which does not exist in other literature. In Section 3 we obtain some properties of non-degenerate states on a general compact quantum group. The last Section 4 is devoted to the positive convolution operators on finite quantum groups, and constructions of operators with similar properties on infinite compact quantum groups by free product.

Let us firstly present some preliminaries and notations on noncommutative $L_{p}$-spaces and free products for later use. All the facts mentioned below are well-known.

Here we recall some basics of noncommutative $L_{p}$-spaces on finite von Neumann algebras. We refer to [Tak02] for the theory of von Neumann algebras and to [PX03] for more information on noncommutative $L_{p}$-spaces. Let $\mathcal{M}$ be a finite von Neumann algebra equipped with a normal faithful tracial state $\tau$. Let $1\leq p<\infty$. For each $x\in\mathcal{M}$, we define

One can show that $\|\|_{p}$ is a norm on $\mathcal{M}$. The completion of $(\mathcal{M},\|\|_{p})$ is denoted by $L_{p}(\mathcal{M},\tau)$ or simply by $L_{p}(\mathcal{M})$. The elements of $L_{p}(\mathcal{M})$ can be described by densely defined closed operators measurable with respect to $(\mathcal{M},\tau)$, as in the commutative case. For convenience, we set $L_{\infty}(\mathcal{M})=\mathcal{M}$ equipped with the operator norm. Since $|\tau(x)|\leq\|x\|_{1}$ for all $x\in\mathcal{M}$, $\tau$ extends to a continuous functional on $L_{1}(\mathcal{M})$. Let $1\leq p,q,r\leq\infty$ be such that $1/p+1/q=1/r$. If $x\in L_{p}(\mathcal{M})$ and $y\in L_{q}(\mathcal{M})$, then $xy\in L_{r}(\mathcal{M})$ and the following Hölder inequality holds:

In particular, if $r=1$, $|\tau(xy)|\leq\|xy\|_{1}\leq\|x\|_{p}\|y\|_{q}$ for arbitrary $x\in L_{p}(\mathcal{M})$ and $y\in L_{q}(\mathcal{M})$. This defines a natural duality between $L_{p}(\mathcal{M})$ and $L_{q}(\mathcal{M})$: $\langle x,y\rangle=\tau(xy)$. For any $1\leq p<\infty$ we have $L_{p}(\mathcal{M})^{*}=L_{q}(\mathcal{M})$ isometrically.

1.2. Free products

We firstly recall some constructions of free product of C*-algebras, for which we refer to [VDN92] and [NS06] for details. Consider a family of unital C*-algebras $(A_{i},\phi_{i})_{i\in I}$ with distinguished faithful states $\phi_{i}$ and associated GNS constructions $(\pi_{i},H_{i})$. Set $\mathring{A_{i}}=\ker\phi_{i}$ and $\mathring{a_{i}}=a_{i}-\phi_{i}(a_{i})1$ for each $i$ and $a_{i}\in A_{i}$. Construct a vector space

We equip $A$ with an algebra structure such that $\mathbf{1}$ is the identity and the multiplication of a letter $a\in\mathring{A_{i}}$ with an elementary tensor $a_{1}\otimes a_{2}\otimes\cdots\otimes a_{n}$ in $\mathring{A}_{i_{1}}\otimes\mathring{A}_{i_{2}}\otimes\cdots\otimes\mathring{A}_{i_{n}}$ is defined as

Moreover, we give an involution on $A$ by

It then can be shown that the algebra $A$ admits a faithful $*$-representation $(\pi,H,\xi)$ such that $\pi|_{A_{i}}=\pi_{i}$ for each $i\in I$ and $\phi(\cdot)\coloneqq\langle\pi(\cdot)\xi,\xi\rangle$ restricted on $A_{i}$ coincides with $\phi_{i}$. Moreover the state $\phi$ is faithful on $A$. Then the reduced C*-algebraic free product of $(A_{i})_{i\in I}$ is the C*-algebra generated by $\pi(A)$ in $B(H)$, i.e., the norm closure of $\pi(A)$ in $B(H)$, denoted by $*_{i\in I}^{c_{0}}A_{i}$; and the state extends to $*_{i\in I}^{c_{0}}A_{i}$, called the free product state of $(\phi_{i})_{i\in I}$ and denoted by $*_{i\in I}\phi_{i}$. If moreover each $A_{i}=\mathcal{M}_{i}$ is a von Neumann algebra and each $\phi_{i}$ is normal, then the weak closure of $\pi(A)$ in $B(H)$, is defined to be the von Neumann algebraic free product of $(\mathcal{M}_{i})_{i\in I}$, denoted by $\bar{*}_{i\in I}\mathcal{M}_{i}$, and the free product state $\phi=*_{i\in I}\phi_{i}$ is also normal. Also, we remark that if each $\phi_{i}$ is a tracial state, then $\phi=*_{i\in I}\phi_{i}$ is also tracial.

Let $A_{i}$ and $B_{i}$ be unital C*-algebras with distinguished faithful states $\phi_{i}$ and $\psi_{i}$ ($i\in I$) respectively, and let $T_{i}:A_{i}\to B_{i}$ be a unital state preserving map for each $i\in I$. Set $(A,\phi)=*_{i\in I}(A_{i},\phi_{i})$ and $(B,\psi)=*_{i\in I}(B_{i},\psi_{i})$. Then it is obvious that

defines a unital state preserving map from the algebraic free products $(A,\phi)$ to $(B,\psi)$. We denote by $T=*_{i\in I}T_{i}$, and call it the free product map of the $T_{i}$. Similarly, we may define the c-free (conditionally free) product state in the sense of Bożejko, Leinert and Speicher [BLS96]. Let $(A_{i},\phi_{i})$ be as above and let $\rho_{i}$ be further states respectively on $A_{i}$ for each $i$. The conditional free product of $(\rho_{i})_{i}$ is the functional $\omega:=*_{(\psi_{i})}\rho_{i}$ on $(A,\phi)=*_{i\in I}(A_{i},\phi_{i})$ defined by the prescription $\omega(1)=1$ and

for all $n\geq 1$, $i(1)\neq\cdots\neq i(n)$ elements in $I$ and $a_{j}\in\ker\phi_{i(j)}$ for $j=1,\ldots,n$. It is shown in [BLS96, Theorem 2.2] that the conditional free product of states is again a state.

Let $A$ be a finite dimensional C*-algebra equipped with a faithful tracial state $\tau$. The associated noncommutative $L_{p}$-spaces will be denoted by $L_{p}(A)$. For a subset $E\subset A$, we denote by $E_{+}$ the positive part of $E$.

Recall that $A$ can be identified with a direct sum of matrix algebras, that is, there exist some finite dimensional Hilbert spaces $H_{1},\ldots,H_{m}$ such that the following $*$-isomorphism holds

We will not distinguish the above two C*-algebras in the sequel. For each $i\in\{1,\ldots,m\}$, let $\xi_{1}^{i},\ldots,\xi_{n_{i}}^{i}$ be an orthonormal basis for $H_{i}$, and define the operator $e_{pq}^{i}\in B(H_{i})$ by $e_{pq}^{i}(v)=\langle v,\xi_{q}^{i}\rangle_{H_{i}}\xi_{p}^{i}$ for all $v\in H_{i}$ and $p,q\in\{1,\ldots,n_{i}\}$. Take any $x=x_{1}\oplus\cdots\oplus x_{m}\in A$ with $x_{i}\in B(H_{i})$ for each $i\in\{1,\ldots,m\}$, and let $\lambda_{1}^{i},\ldots,\lambda_{n_{i}}^{i}$ be the eigenvalues of $|x_{i}|\in B(H_{i})$ ($1\leq i\leq m$) ranged in non-increasing order and counted according to multiplicity. We can find a direct sum of unitaries $u=u_{1}\oplus u_{2}\oplus\cdots\oplus u_{m}$ with $u_{i}\in B(H_{i})$ for each $i$ such that $|x_{i}|(u_{i}\xi_{k}^{i})=\lambda_{k}^{i}(u_{i}\xi_{k}^{i})$ for all $k\in\{1,\ldots,n_{i}\}$ and $i\in\{1,\ldots,m\}$, that is, $u^{*}|x|u=\sum_{i}\sum_{k=1}^{n_{i}}\lambda_{k}^{i}e_{kk}^{i}$. If we write $\beta_{k}^{i}=\tau(e_{kk}^{i})\in$ for $k\in\{1,\ldots,n_{i}\}$ and $i\in\{1,\ldots,m\}$, then the $L_{p}$-norm of $x$ for $1\leq p<\infty$ is

We will prove in this section the result below.

Let $A$ be a finite dimensional C*-algebra equipped with a faithful tracial state $\tau$, and $T:A\to A$ be a unital $2$-positive trace preserving map on $A$. Then

Equivalently we can rewrite the above condition (1.3) as

Recall that the $L_{2}$-norms assert some differential properties. The following lemma is elementary.

Let $A$ be a C*-algebra with a state $\varphi$ and $T:A\to A$ be a positive map on $A$. Let $O\subset A_{h}$ be an open set in the space $A_{h}$ of all selfadjoint elements in $A$. The function $f:O\ni x\mapsto\varphi((Tx)^{2})$ is infinitely (Fréchet) differentiable in $O$ and for $x\in O$, $f^{\prime}(x)=\varphi(TxT\cdot)+\varphi(T\cdot Tx)$, $f^{\prime\prime}\equiv 2\varphi(T\cdot T\cdot)$, $f^{(n)}\equiv 0$, $n\geq 3$.

In general a norm estimate can be reduced to the argument on positive cones.

Assume firstly $1\leq p<2$ and $\|Tx\|_{2}\leq\|x\|_{p}$ for all $x\in A$. Note that $\|Tx\|_{2}=\||T|x\|_{2}$. Observe that $T^{*}$ is also a positive trace preserving map on $A$, and hence so is $|T|$. We choose an element $x\in A$ such that $\tau(x)=0$ and $|T|x=\lambda x$, with

Then taking the second derivative at $\varepsilon=0$ we get $\lambda^{2}\leq(p-1)<1$, as desired.

Now we suppose (1.3) holds. Set \mathring{A}=\{x\in A\bigm{|}\tau(x)=0\} and take \sigma=\{x\in A_{+}\bigm{|}\tau(x)=1\}=(1+\mathring{A})_{+} which is exactly the set of positive elements in the unit sphere of $L_{1}(A)$. We first show that there exists $1\leq p<2$ and a neighborhood $U$ of $1$ such that

Using the previous lemma we see that $F$ is infinitely differentiable at any $x\in A_{+}\setminus\{0\}$ and

Since $T$ is unital and preserves the trace, it follows that for $y\in\mathring{A}$,

Then consider the second order Taylor expansion of $F$ at $1$. We can find a $\delta_{1}>0$ such that for all $\|y\|_{2}\leq\delta_{1}$, $y\in\mathring{A}$, we have $1+y\in A_{+}$ and

Recall that by (1.3), $\|Ty\|_{2}^{2}-\|y\|_{2}^{2}<0$ for $y\in\mathring{A}$. Thus by continuity,

Since the function $y\mapsto\|Ty\|_{2}^{2}-\|y\|_{2}^{2}$ is $2$-homogeneous, we get

Take $\delta_{0}\in(0,\delta_{1})$ such that

Then for $y\in\mathring{A}$, $\|y\|_{2}\leq\delta_{0},$

This, together with $(**)$, implies that, putting $\lambda=(\lambda_{1},\ldots,\lambda_{K})$,

for all $p\geq 2-\frac{|c|}{8}\coloneqq p_{1}$. So U=\{1+y\bigm{|}y=y^{*}\in A,\|y\|_{2}<\delta\} is the desired neighborhood in (1.4).

Now we can derive the inequality for all $x\in\sigma$. For $x\in\sigma\setminus U\subset(1+\mathring{A})_{+}\setminus\{1\}$, we write $x=1+y$ with $y\in\mathring{A}\backslash\{0\}$ and then by (1.3) and the trace preserving property we have $\|Tx\|_{2}^{2}=1+\|Ty\|_{2}^{2}<1+\|y\|_{2}^{2}=\|x\|_{2}^{2}$. Note also that $\sigma$ is compact, so we can find $M<1$ such that $\|Tx\|_{2}/\|x\|_{2}<M$ for all $x\in\sigma\backslash U$. Given $p<2$, let $C_{p}$ be the optimal constant for the inequality $\|x\|_{2}\leq C_{p}\|x\|_{p}$ for $x\in A$, then $C_{p}\to 1$ when $p\to 2$. Take $p_{0}\geq p_{1}$ such that $C_{p_{0}}\leq M^{-1}$. We get then

As a result, for all $p\in[p_{0},2]$, it holds that

Since the norm is homogeneous and $T$ is $2$-positive, the above inequality holds for all $x\in A$ as well. ∎

Apart from the above elementary proof, we would like to give an alternative simpler approach which yields a little bit stronger conclusion. The argument, however, depends heavily on the following recent and deep result on the convexity of $L_{p}$-spaces:

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful semifinite normal trace $\phi$. Let $\mathcal{N}$ be a von Neumann subalgebra such that the restriction of $\phi$ to $\mathcal{N}$ is semifinite. Denote by $\mathcal{E}$ the unique $\phi$-preserving conditional expectation from $\mathcal{M}$ onto $\mathcal{N}$. For $1<p\leq 2$, we have

For $2<p<\infty$, the inequality is reversed.

Immediately we may deduce Theorem 1.1 as follows. Note that the result below is slightly stronger than the statement of Theorem 1.1.

Let $A$ be a finite dimensional C*-algebra equipped with a faithful tracial state $\tau$, and $T:A\to A$ be a unital trace preserving map on $A$. Then

Moreover, if the above assertions are satisfied, then

The necessity has been already proved in the proof of Theorem 1.1. Now, assume $\lambda<1$. Let $x\in A$ and $y=x-\tau(x)1$. Write $a=\tau(x)$. Since $T$ is trace preserving, $\tau(Ty)=\tau(y)=0$. For $p\leq 2$ we denote by $c_{p}$ the best constant with $\|\cdot\|_{2}\leq c_{p}\|\cdot\|_{p}$. Then $(p-1)/c_{p}^{2}\to 1$ when $p\to 2$. Take $p<2$ such that $(p-1)/c_{p}^{2}>\lambda^{2}$, then we have

Let $A$ be a finite dimensional C*-algebra equipped with a faithful tracial state $\tau$, and $T:A\to A$ be a unital trace preserving map on $A$. Consider the restriction of $T$ on the subspace $\{x\in A:\tau(x)=0\}$ of $A$ and its adjoint, then we see that

Then the above theorem also implies that if there exists $1<p<2$ such that

and equivalently for $2<q<\infty$ with $1/p+1/q=1$,

It is easy to see that the free product of unital trace preserving completely positive maps can be extended to the $L_{p}$-spaces on using the interpolation between $L_{1}$ and $L_{\infty}$. But in general it is a delicate problem for the extension of algebraic free product of unital trace preserving maps onto the associated $L_{p}$-spaces. Here we provide a method to construct unital trace preserving $L_{p}$-improving operators on the free product of finite-dimensional C*-algebras. To see this we need the following trivial claim.

which gives the claim for $q=\infty$. The inequality for $2\leq q\leq\infty$ then follows from the Hölder inequality.∎

Let $(A_{i},\tau_{i})$, $1\leq i\leq n$ be a finite family of finite dimensional C*-algebras and set $(\mathcal{A},\tau)=\bar{*}_{1\leq i\leq n}(A_{i},\tau_{i})$ to be the von Neumann algebraic free product. For each $1\leq i\leq n$, $T_{i}$ is a unital trace preserving map such that

for some $1<p<2$. Then the (algebraic) free product map $T=*_{1\leq i\leq n}T_{i}$ on $*_{1\leq i\leq n}A_{i}$ extends to a map such that

Consider $R=T^{*}$ and $R_{i}=T_{i}^{*}$ for all $1\leq i\leq n$, then $R=R_{1}*\cdots*R_{n}$. By density, consider $x\in*_{1\leq i\leq n}(A_{i},\tau_{i})$ in the algebraic free product and we will show that

for some $q>2$ independent of the choice of $x$. Now fix some $r\geq 1$. For each $i$, choose a family $(e_{k}^{(i)})_{k=1}^{n_{i}}$ of eigenvectors of $|R_{i}|$ which forms an orthonormal basis of $\mathring{A}_{i}$ under $\tau_{i}$, then $E_{r}=\{e_{\underline{k}}^{\underline{i}}=e_{k_{1}}^{(i_{1})}\cdots e_{k_{r}}^{(i_{r})}:1\leq k_{j}\leq n_{j},1\leq j\leq r,i_{1}\neq\cdots\neq i_{r}\}$ forms an orthonormal basis of $\oplus_{i_{1}\neq\cdots\neq i_{r}}\mathring{A}_{i_{1}}\otimes\cdots\otimes\mathring{A}_{i_{r}}$ which are also eigenvectors of $|R|$. Note that $|E_{r}|\leq n^{r}m^{r}$ for $m=\max_{j}n_{j}$. Write additionally $c=\max_{k,i}\|e_{k}^{(i)}\|_{\infty}^{2}$. Then for any $y_{r}\in\oplus_{i_{1}\neq\cdots\neq i_{r}}\mathring{A}_{i_{1}}\otimes\cdots\otimes\mathring{A}_{i_{r}}$ the above claim yields

Write $x=\tau(x)1+\sum_{r\geq 1}x_{r}$ where $x_{r}\in\oplus_{i_{1}\neq\cdots\neq i_{r}}\mathring{A}_{i_{1}}\otimes\cdots\otimes\mathring{A}_{i_{r}}$. Note that $\|Rx_{r}\|_{2}\leq\lambda^{r}\|x_{r}\|_{2}$ according to (1.6) and the choice of $E_{r}$. Together with Theorem 1.5 and (1.7),

Observe that $(q-1)(cnm)^{\frac{1}{2}-\frac{1}{q}}$ tends to $1$ whenever $q\to 2$ and that $\lambda<1$, so we may choose $2<q<\infty$ such that $\lambda(cnm)^{\frac{1}{2}-\frac{1}{q}}\leq(q-1)^{-1}$. For such a $q$ we then have

Take $1<p^{\prime}<2$ such that $1/p^{\prime}+1/q=1$. Then we get $\|T:L_{p^{\prime}}(\mathcal{A})\to L_{2}(\mathcal{A})\|=1$. ∎

Preliminaries on quantum groups with Fourier analysis

In this section we will do some preparations for discussing convolution operators in the quantum group framework. We will start with some preliminaries on compact quantum groups and then introduce the Fourier series in this setting.

In this short paragraph we recall some basic definitions and properties of compact quantum groups. All proofs of the facts mentioned below without references can be found in [Wor98] and [MVD98].

Consider a unital C*-algebra $A$ and a unital $*$-homomorphism $\Delta:A\to A\otimes A$ called comultiplication on $A$ such that $(\Delta\otimes\iota)\Delta=(\iota\otimes\Delta)\Delta$ and

The following fact due to Woronowicz is fundamental in the quantum group theory.

We will use the Sweedler notation for the comultiplication of an element $a\in A$, i.e. omit the summation and the index in the formula $\Delta(a)=\sum_{i}a_{(1),i}\otimes a_{(2),i}$ and write simply $\Delta(a)=\sum a_{(1)}\otimes a_{(2)}$.

Finally we turn to the dual free product of compact quantum groups. The following construction is given by [Wan95].

2. Fourier analysis

where $\mathcal{E_{\alpha}}x\coloneqq\sum_{i,j}x_{ij}^{\alpha}u_{ij}^{\alpha}$ is the orthogonal projection of $x$ onto $E_{\alpha}$.

Combining the last equality with (2.7) proves the desired (2.6).

(1) Let $G$ be a compact group and define

In particular for $f\in L_{2}(G)$, we have

and we have the Fourier expansion and the Plancherel formula

Observe that the multiplier $m_{a}$ (or $m_{a}^{\prime}$ resp.) is unital, i.e. $m_{a}(1)=1$ ($m_{a}^{\prime}(1)=1$ resp.) if and only if $a_{1}=1$.

Then by (2.1), a straightforward calculation shows that

Non-degenerate states and applications to Hopf images

In this short section we give the key lemma on non-degenerate states, which will be of use for our main results. We need the following observation adapted from [Wor98, Lemma 2.1]. The result is mentioned in [Soł05].

Then $\mathcal{I}$ is a closed left ideal of $A\otimes A$. Define

Since $\Psi_{L}$ is a difference of two unital completely positive maps, we see that $\Psi_{L}$ is a completely bounded map with norm at most $2$. We will prove that

In fact, given $x\in A$, by the coassociativity of $\Delta$ we have

where by the convolution invariance assumption and the coassociativity of $\Delta$ we have

Note that $q_{1}=q_{2}=q_{3}=q_{4}$. So $(\rho_{i}\otimes\rho)(q^{*}q)=0$ and $(\Psi_{L}\otimes\iota)\Delta(A)\subset\mathcal{I}$ is proved.

Now by the density of $(1\otimes A)\Delta(A)$ in $A\otimes A$ and the complete boundedness of $\Psi_{L}$, it follows that $\Psi_{L}(A)\otimes 1\subset\overline{(1\otimes A)(\Psi_{L}\otimes\iota)\Delta(A)}$ is also contained in the closed left ideal $\mathcal{I}$, which means that for any $i\in I$ and $x\in A$,

Recall that $(\rho_{i})_{i\in I}$ separates the points of $A_{+}$, so we have $\Psi_{L}(x)=0$ and $(\rho\otimes\iota)\Delta(x)=\rho(x)1$ for all $x\in A$.

A similar argument applies as well to the map $\Psi_{R}(x)=(\iota\otimes\rho)\Delta(x)-\rho(x)1$, $x\in A$. So $\rho=h$ is the Haar state.∎

We immediately obtain the following fact.

Main results

In this section we aim to give several characterizations of $L_{p}$-improving convolutions given by states on finite quantum groups, and also give the constructions for the free product of finite quantum groups. We will start with some discussions on multipliers on compact quantum groups. In this section we keep the notation of multipliers $m_{a}$, $m_{a}^{\prime}$ and convolutions $\varphi_{1}\star\varphi_{2}$ given in Section 2.

whenever $n\to\infty$. And for $\alpha=1$,

Without loss of generality we only discuss the left multiplier $m_{a}$ and prove the equivalence (1)$\Leftrightarrow$(3)$\Leftrightarrow$(4).

The equivalence between (3) and (4) was proved in the previous lemma. Therefore the theorem is established. ∎

Now with these remarks and Lemma 3.3 in hand, we may reformulate Theorem 4.3 in terms of convolution operators using the above arguments.

$\lim_{n}\frac{1}{n}\sum_{k=1}^{n}\psi^{\star k}=h$;

Let $G$ be a finite group and $\mu$ be a probability measure on $G$. Then there is a $1\leq p<2$ such that

so the convolution operators associated to $\Phi$ are just the Fourier-Schur multiplier on $\Gamma$ associated to $\varphi$. Our preceding argument in particular yields the following result extending [Rit84, Theorem 2(a)].

Let $\Gamma$ be a finite group and $\varphi$ be a positive definite function on $\Gamma$ with $\varphi(e)=1$. Let $M_{\varphi}$ be the associated Fourier-Schur multiplier operator determined by $M_{\varphi}(\lambda(\gamma))=\varphi(\gamma)\lambda(\gamma)$ for all $\gamma\in\Gamma$. Then there exists $1\leq p<2$ such that

if and only if $|\varphi(\gamma)|<1$ for any $\gamma\in\Gamma\setminus\{e\}$.

which yields an impossible equivalence between the norms $\|\cdot\|_{2}$ and $\|\cdot\|_{p}$.

where we have used the fact that the comultiplication $\Delta$ is an homomorphism. Then the equality (4.2) follows from a standard density argument. Now taking in Theorem 1.9 each $T_{i}$ to be a convolution operator on a finite quantum group, we get the following corollary:

The author is indebted to his advisors Quanhua Xu and Adam Skalski for their helpful discussions and constant encouragement, and to Professor Gilles Pisier for his careful reading and pointing out a mistake in the preprint version. The author also thanks the referee for a careful reading of the manuscript and useful suggestions. This research was partially supported by the NCN (National Centre of Science), grant no. 2014/14/E/ST1/00525.

References