2 Preliminaries

The class of locally compact quantum groups was first introduced and studied by Kustermans and Vaes [Reference Kustermans and Vaes8, Reference Kustermans and Vaes9]. Recall that a (von Neumann algebraic) locally compact quantum group is a quadruple ${\mathbb G}=(L^{\infty }({\mathbb G}), \Delta , \phi , \psi )$ , where $L^{\infty }({\mathbb G})$ is a von Neumann algebra with identity element $1$ and a co-multiplication $ \Delta : L^{\infty }({\mathbb G})\rightarrow L^{\infty }({\mathbb G})\bar {\otimes }L^{\infty }({\mathbb G}). $ Moreover, $\phi $ and $\psi $ are normal faithful semifinite left and right Haar weights on $L^{\infty }({\mathbb G})$ , respectively. Here, $\bar {\otimes }$ denotes the von Neumann algebra tensor product.

The predual of $L^{\infty }({\mathbb G})$ is denoted by $L^1({\mathbb G})$ which is called quantum group algebra of $\mathbb {G}$ . Then the pre-adjoint of the co-multiplication $\Delta $ induces on $L^1({\mathbb G})$ an associative completely contractive multiplication $\Delta _*:L^1({\mathbb G})\widehat {\otimes }L^1({\mathbb G})\rightarrow L^1({\mathbb G})$ , where $\widehat {\otimes }$ is the operator space projective tensor product. Therefore, $L^1({\mathbb G})$ is a Banach algebra under the product $*$ given by $f*g:=\Delta _*(f\otimes g)\in L^1({\mathbb G})$ for all $f,g\in L^1({\mathbb G})$ . Moreover, the module actions of $L^1({\mathbb G})$ on $L^{\infty }({\mathbb G})$ are given by

$$ \begin{align*} f\cdot x:=(\mathrm{id}\otimes f)(\Delta(x)),\quad x\cdot f:=(f\otimes\mathrm{ id})(\Delta(x)) \end{align*} $$

for all $f\in L^1({\mathbb G})$ and $x\in L^{\infty }({\mathbb G})$ .

For every locally compact quantum group $\mathbb {G}$ , there is a left fundamental unitary operator $W\in L^{\infty }(\mathbb {G})\bar {\otimes } L^{\infty }(\widehat {\mathbb {G}})$ and a right fundamental unitary operator $V\in L^{\infty }(\widehat {\mathbb {G}})'\bar {\otimes } L^{\infty }(\mathbb {G}) $ which the co-multiplication $\Delta $ can be given in terms of W and V by the formula

$$ \begin{align*}\Delta(x)=W^{*}(1 \otimes x) W=V(x \otimes 1) V^{*} \quad\left(x \in L^{\infty}(\mathbb{G})\right), \end{align*} $$

where $ L^{\infty }(\widehat {\mathbb G}):={\{(f\otimes \mathrm {id})(W)\;:\; f\in L^1(\mathbb G)\}}^{"} $ . The Gelfand–Naimark–Segal (GNS) representation space for the left Haar weight will be denoted by $L^2({\mathbb G})$ . Put $\widehat {W}=\sigma W^*\sigma $ , where $\sigma $ denotes the flip operator on $B(L^2({\mathbb G})\otimes L^2({\mathbb G}))$ , and define

$$ \begin{align*} \widehat{\Delta}:L^{\infty}(\widehat{\mathbb G})\rightarrow L^{\infty}(\widehat{\mathbb G})\bar{\otimes}L^{\infty}(\widehat{\mathbb G}),\;\;\; x\mapsto\widehat{W}^*(1\otimes x)\widehat{W}, \end{align*} $$

which is a co-multiplication. One can also define a left Haar weight $\hat {\varphi }$ and a right Haar weight $\hat {\psi }$ on $L^{\infty }(\widehat {\mathbb G})$ that $\widehat {\mathbb G}=(L^{\infty }(\widehat {\mathbb G}),\widehat {\Gamma }, \hat {\varphi }, \hat {\psi }),$ the dual quantum group of ${\mathbb G}$ , turn it into a locally compact quantum group. Moreover, a Pontryagin duality theorem holds, that is, $\widehat {\widehat {\mathbb G}}={\mathbb G}$ (for more details, see [Reference Kustermans and Vaes8, Reference Kustermans and Vaes9]). The reduced quantum group $C^*$ -algebra of $L^{\infty }({\mathbb G})$ is defined as

$$ \begin{align*} C_0({\mathbb G}):=\overline{\{(\mathrm{id}\otimes\omega)(W);\ \omega\in B(L^2({\mathbb G}))_*\}}^{\|.\|}. \end{align*} $$

We say that ${\mathbb G}$ is compact if $C_0({\mathbb G})$ is a unital $C^*$ -algebra. The co-multiplication $\Delta $ maps $C_0({\mathbb G})$ into the multiplier algebra $M(C_0({\mathbb G})\otimes C_0({\mathbb G}))$ of the minimal $C^*$ -algebra tensor product $C_0({\mathbb G})\otimes C_0({\mathbb G})$ . Thus, we can define the completely contractive product $*$ on $C_0({\mathbb G})^*=M({\mathbb G})$ by

$$ \begin{align*} \langle \omega*\nu, x\rangle=(\omega\otimes\nu)(\Delta x)\quad (x\in C_0({\mathbb G}), \omega,\nu\in M({\mathbb G})), \end{align*} $$

whence $(M({\mathbb G}), *)$ is a completely contractive Banach algebra and contains $L^1({\mathbb G})$ as a norm closed two-sided ideal. If X is a Banach right $L^1({\mathbb G})$ -submodule of $L^{\infty }({\mathbb G})$ with $1\in X$ , then a left invariant mean on X, is a functional $m\in X^*$ satisfying

$$ \begin{align*} \|m\|=\langle m, 1\rangle=1, \quad \langle m, x \cdot f\rangle=\langle f, 1\rangle \langle m, x\rangle\quad (f\in L^1({\mathbb G}), x\in X). \end{align*} $$

Right and (two-sided) invariant means are defined similarly. A locally compact quantum group ${\mathbb G}$ is said to be amenable if there exists a left (equivalently, right, or two-sided) invariant mean on $L^{\infty }({\mathbb G})$ (see [Reference Desmedt, Quaegebeur and Vaes4, Proposition 3]). A standard argument, used in the proof of [Reference Lau10, Theorem 4.1] on Lau algebras shows that ${\mathbb G}$ is amenable if and only if $L^1({\mathbb G})$ is left $1$ -amenable. We also recall that, $\mathbb {G}$ is called co-amenable if $L^1({\mathbb G})$ has a bounded approximate identity.

The right fundamental unitary V of $\mathbb {G}$ induces a co-associative co-multiplication

$$ \begin{align*} \Delta^{r}: \mathcal{B}\left(L^{2}(\mathbb{G})\right) \ni x \mapsto V(x \otimes 1) V^{*} \in \mathcal{B}\left(L^{2}(\mathbb{G})\right) \bar{\otimes} \mathcal{B}\left(L^{2}(\mathbb{G})\right), \end{align*} $$

and the restriction of $\Delta ^{r}$ to $L^{\infty }(\mathbb {G})$ yields the original co-multiplication $\Delta $ on $L^{\infty }(\mathbb {G})$ . The pre-adjoint of $\Delta ^{r}$ induces an associative completely contractive multiplication on space $\mathcal {T}\left (L^{2}(\mathbb {G})\right )$ of trace class operators on $L^{2}(\mathbb {G})$ , defined by

$$ \begin{align*} \triangleright: \mathcal{T}\left(L^{2}(\mathbb{G})\right) \widehat{\otimes} \mathcal{T}\left(L^{2}(\mathbb{G})\right) \ni \omega \otimes \tau \mapsto \omega \triangleright \tau=\Delta_{*}^{r}(\omega \otimes \tau) \in \mathcal{T}\left(L^{2}(\mathbb{G})\right), \end{align*} $$

where $\widehat {\otimes }$ denotes the operator space projective tensor product.

It was shown in [Reference Hu, Neufang and Ruan6, Lemma 5.2], that the pre-annihilator $L^{\infty }(\mathbb {G})_{\perp }$ of $L^{\infty }(\mathbb {G})$ in $\mathcal {T}\left (L^{2}(\mathbb {G})\right )$ is a norm closed two-sided ideal in $\left (\mathcal {T}\left (L^{2}(\mathbb {G})\right ), \triangleright \right )$ and the complete quotient map

$$ \begin{align*} \pi: \mathcal{T}\left(L^{2}(\mathbb{G})\right) \ni \omega \mapsto f=\left.\omega\right|{}_{L^{\infty}(\mathbb{G})} \in L^{1}(\mathbb{G}) \end{align*} $$

is a completely contractive algebra homomorphism from $\mathcal {T}_{\triangleright }({\mathbb G}):=\left (\mathcal {T}\left (L^{2}(\mathbb {G})\right ), \triangleright \right )$ onto $L^{1}(\mathbb {G})$ . The multiplication $\triangleright $ defines a canonical $\mathcal {T}_{\triangleright }({\mathbb G})$ -bimodule structure on $\mathcal {B}\left (L^{2}(\mathbb {G})\right )$ . Note that since $V \in L^{\infty }(\widehat {\mathbb {G}}^{\prime }) \bar {\otimes } L^{\infty }(\mathbb {G})$ , the bimodule action on $L^{\infty }(\widehat {\mathbb {G}})$ becomes rather trivial. Indeed, for $\hat {x} \in L^{\infty }(\widehat {\mathbb {G}})$ and $\omega \in \mathcal {T}_{\triangleright }({\mathbb G}),$ we have

$$ \begin{align*} \hat{x} \triangleright \omega=(\omega \otimes \iota) V(\hat{x} \otimes 1) V^{*}=\langle\omega, \hat{x}\rangle 1, \quad \omega \triangleright \hat{x}=(\iota \otimes \omega) V(\hat{x} \otimes 1) V^{*}=\langle\omega, 1\rangle \hat{x}. \end{align*} $$

This implies that $L^{\infty }(\widehat {\mathbb {G}})\subseteq \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ . It is also known from [Reference Hu, Neufang and Ruan6, Proposition 5.3] that $B(L^2({\mathbb G}))\triangleright \mathcal {T}_{\triangleright }({\mathbb G})\subseteq L^{\infty }({\mathbb G}).$ In particular, the actions of $\mathcal {T}_{\triangleright }({\mathbb G})$ on $L^{\infty }(\mathbb {G})$ satisfies

$$ \begin{align*} \omega \triangleright x=\pi(\omega)\cdot x, \quad x\triangleright \omega=x\cdot\pi(\omega) \end{align*} $$

for all $\omega \in \mathcal {T}_{\triangleright }({\mathbb G})$ and $ x\in L^{\infty }(\mathbb {G})$ .

3 Invariant means on weakly almost periodic functionals

Let I be a closed ideal of the Banach algebra $\mathcal {A}$ . Then for every $b \in I$ and $x \in I^{*}$ , define $x \bullet b, b \bullet x \in \mathcal {A}^*$ as follows:

$$ \begin{align*} \langle x \bullet b, a\rangle=\langle x, b a\rangle, \quad\langle b\bullet x, a\rangle=\langle x, a b\rangle \quad\left(a \in \mathcal{A}\right). \end{align*} $$

We note that, given $a\in {\mathcal A}, b_1, b_2\in I$ , and $x\in I^*$ , for $a'\in {\mathcal A}$ , we have

$$ \begin{align*} \langle a\cdot((b_1\cdot x)\bullet b_2), a'\rangle &=\langle (b_1\cdot x)\bullet b_2, a'a\rangle =\langle b_1\cdot x, b_2a'a\rangle =\langle x, b_2a'ab_1\rangle\\ &= \langle ab_1\cdot x, b_2a'\rangle =\langle (ab_1\cdot x)\bullet b_2, a'\rangle, \end{align*} $$

so that, $ a\cdot ((b_1\cdot x)\bullet b_2)=(ab_1\cdot x)\bullet b_2$ .

Lemma 3.1 Let ${\mathcal A}$ be a Banach algebra, and let I be a closed ideal of ${\mathcal A}$ with a bounded approximate identity. Then

$$ \begin{align*} \mathrm{WAP}(I)\bullet I\subseteq \mathrm{WAP}(\mathcal{A}),\quad I\bullet\mathrm{WAP}(I)\subseteq \mathrm{WAP}(\mathcal{A}).\end{align*} $$

Proof Let $x\in \mathrm {WAP}(I)$ and $b_1, b_2\in I$ . Suppose that $(a_n)$ is a bounded sequence in ${\mathcal A}$ . Then $(a_nb_1)$ is a bounded sequence in I and so by weak compactness of the map $\lambda _x: I\rightarrow I^*$ , there is a subsequence $(a_{n_j}b_1)$ of $(a_nb_1)$ such that $(a_{n_j}b_1\cdot x)$ converges weakly in $I^*$ to some $y\in I^*$ . Now, for each $m\in {\mathcal A}^{**}$ , define the functional $b_2\bullet m\in I^{**}$ as follows:

$$ \begin{align*} \langle b_2\bullet m, z\rangle=\langle m, z\bullet b_2\rangle\quad (z\in I^*). \end{align*} $$

It follows that

$$ \begin{align*} \langle m,a_{n_j}\cdot((b_1\cdot x)\bullet b_2)\rangle &= \langle m,(a_{n_j}b_1\cdot x)\bullet b_2\rangle \\ &= \langle b_2\bullet m, a_{n_j}b_1\cdot x\rangle \rightarrow \langle b_2\bullet m, y\rangle\\ &= \langle m, y\bullet b_2\rangle \end{align*} $$

for all $m\in {\mathcal A}^{**}$ . That is, $(b_1\cdot x)\bullet b_2\in \mathrm {WAP}(\mathcal {A})$ . Since I has a bounded right approximate identity, it follows from [Reference Dales and Lau2, Proposition 3.12] that $I\cdot \mathrm {WAP}(I)=\mathrm {WAP}(I)$ . This shows that $\mathrm {WAP}(I)\bullet I\subseteq \mathrm {WAP}(\mathcal {A})$ . The inclusion $I\bullet \mathrm {WAP}(I)\subseteq \mathrm {WAP}(\mathcal {A})$ can be proved similarly.

Proof We only prove the right version of the theorem. Similar arguments will establish the left side version.

(i) $\Rightarrow $ (ii). Let m be a right invariant $\varphi |_I$ -mean on $\mathrm {WAP}(I)$ . This means that for every $x\in \mathrm {WAP}(I)$ and $b\in I,$ we have

$$ \begin{align*} \langle m, b\cdot x\rangle=\varphi(b) \langle m , x\rangle. \end{align*} $$

We denote by $\imath :I\rightarrow {\mathcal A}$ the canonical embedding map. By [Reference Young18, Corollary to Lemma 1], the map $R:=\imath ^*: {\mathcal A}^*\rightarrow I^*$ maps $\operatorname {WAP}({\mathcal A})$ to $\mathrm {WAP}(I)$ . Define ${\widetilde m}:=m\circ R\in {\mathcal A}^{**}$ . It is easy to see that $ \langle {\widetilde m}, \varphi \rangle =1. $ Let $(e_{\alpha })$ be a bounded approximate identity for I. By [Reference Dales and Lau2, Proposition 3.12], we have $I\cdot \operatorname {WAP}(I)=\operatorname {WAP}(I)\cdot I=\operatorname {WAP}(I)$ . Thus, $\lim _{\alpha } e_{\alpha }\cdot R(y)=R(y)$ for all $y\in \operatorname {WAP}({\mathcal A})$ . Moreover, by [Reference Runde14, Proposition 2.1.6], $\mathrm {WAP}(I)$ becomes a Banach ${\mathcal A}$ -bimodule and since I is an ideal in ${\mathcal A}$ , it is not hard check that $R(a\cdot y)=a\cdot R(y)$ for all $a\in {\mathcal A}$ and $y\in \operatorname {WAP}({\mathcal A})$ . Therefore, for every $a\in {\mathcal A}$ and $y\in \operatorname {WAP}({\mathcal A})$ , we have

$$ \begin{align*} \langle {\widetilde m}, a\cdot y\rangle&=\langle m,R(a\cdot y)\rangle =\langle m,a\cdot R(y)\rangle\\ &=\lim_{\alpha}\langle m,a\cdot (e_{\alpha}\cdot R(y))\rangle =\lim_{\alpha}\langle m, ae_{\alpha}\cdot R(y)\rangle\\ &=\lim_{\alpha}\varphi(ae_{\alpha})\langle m, R(y)\rangle =\varphi(a)\varphi(e_{\alpha})\langle {\widetilde m}, y\rangle=\varphi(a)\langle {\widetilde m}, y\rangle. \end{align*} $$

Thus, ${\widetilde m}$ is a right invariant $\varphi $ -mean on $\operatorname {WAP}({\mathcal A})$ .

(ii) $\Rightarrow $ (i). Let $m \in {\mathcal A}^{**}$ be a right invariant $\varphi $ -mean on $\operatorname {WAP}({\mathcal A})$ . Fix $b_{0} \in I$ with $\varphi (b_0)=1$ . Since $\operatorname {WAP}(I)\bullet b_0\subseteq \operatorname {WAP}({\mathcal A})$ , by Lemma 3.1, we can define $\tilde {m} \in \operatorname {WAP}(I)^{*}$ as follows:

$$ \begin{align*} \langle\tilde{m}, x\rangle=\left\langle m, x \bullet b_{0}\right\rangle \quad\left(x \in \operatorname{WAP}(I)\right). \end{align*} $$

It is easily verified that

$$ \begin{align*} \langle\tilde{m},\varphi|_{I}\rangle=\langle m, \varphi|_{I}\bullet b_{0} \rangle=\langle m, \varphi\rangle=1. \end{align*} $$

Moreover, for every $b\in I$ and $x \in \operatorname {WAP}(I)$ , we have

$$ \begin{align*} \begin{aligned} \langle\tilde{m}, b \cdot x\rangle =\left\langle m, (b\cdot x)\bullet b_{0}\right\rangle &= \left\langle m, b\cdot(x\bullet b_0) \right\rangle \\ &=\left.\varphi\right|{}_{I}\left(b\right)\left\langle m, x \bullet b_0\right\rangle \\ &=\left.\varphi\right|{}_{I}\left(b\right)\langle\tilde{m}, x\rangle. \end{aligned} \end{align*} $$

Therefore, $\tilde {m}$ is a right $\left .\varphi \right |{}_{I}$ -mean on $\operatorname {WAP}(I)$ .

We now consider some special cases. Suppose that ${\mathbb G}$ is a locally compact quantum group. Then ${\Bbb G}$ has a canonical co-involution ${\mathcal R}$ , called the unitary antipode of ${\Bbb G}$ . That is, ${\mathcal R}: L^{\infty }({\Bbb G})\longrightarrow L^{\infty }({\Bbb G})$ is a $^*$ -anti-homomorphism satisfying ${\mathcal R}^2=\mathrm {id}$ and $ \Delta \circ {\mathcal R}=\sigma ({\mathcal R}\otimes {\mathcal R})\circ \Delta , $ where $\sigma $ is the flip map on $L^2({\Bbb G})\otimes L^2({\Bbb G})$ . Then ${\mathcal R}$ induces a completely isometric involution on $L^1({\Bbb G})$ defined by

$$ \begin{align*}\langle x, f'\rangle=\overline{\langle f, {\mathcal R}(x^*)\rangle}\quad(x\in L^{\infty}({\Bbb G}), f\in L^1({\Bbb G})). \end{align*} $$

Hence, $L^1({\Bbb G})$ becomes an involutive Banach algebra.

Now, assume that m is a left (resp. right) invariant $1$ -mean on $\operatorname {WAP}(L^1({\mathbb G}))$ , and let $\widetilde {m}\in L^{\infty }({\mathbb G})^*$ be a Hahn–Banach extension of m. It is not hard to see that $n:=\widetilde {m}^{\circ }|_{\operatorname {WAP}(L^1({\mathbb G})}$ is a right (resp. left) invariant $1$ -mean on $\operatorname {WAP}(L^1({\mathbb G}))$ , where $\circ : L^{\infty }({\mathbb G})^*\rightarrow L^{\infty }({\mathbb G})^*, m\mapsto m^{\circ }$ is the unique weak $^*$ -weak $^*$ continuous extension of the involution on $L^1({\mathbb G})$ which is called the linear involution (see [Reference Dales and Lau2, Chapter 2, p. 18]. Thus, by Remark 3.3, we obtain that any left (resp. right) invariant $1$ -mean on $\operatorname {WAP}(L^1({\mathbb G}))$ is unique and (two-sided) invariant.

Our next result yields a generalization of [Reference Neufang12, Theorem 2.3] which is concerned with the group algebra $L^1(G)$ as an ideal in the measure algebra $M(G)$ , for a locally compact group G.

Proof To see this, first note that, since we can identify $I^{**}$ with $I^{\perp \perp }$ , it follows that $I^{**}$ is a closed ideal in ${\mathcal A}^{**}$ (see [Reference Dales and Lau2, p. 17]). Fix $b_0\in I$ with $\varphi (b_0)=1$ . Now, suppose that $m\in {\mathcal A}^{**}$ is a right invariant $\varphi $ -mean on ${\mathcal A}^*$ . Since $I^{**}$ is an ideal in ${\mathcal A}^{**}$ , we obtain that $b_0\square m\in I^{**}$ . Furthermore, $\langle b_0\square m, \varphi \rangle =1$ and

$$ \begin{align*} (b_0\square m)\square b=\varphi(b)b_0\square m \end{align*} $$

for all $b\in I$ . Thus, $b_0\square m$ is a right invariant $\varphi |_I$ -mean on ${I}^*$ . For the converse, suppose that $m\in {I}^{**}$ is a right invariant $\varphi |_I$ -mean on $I^*$ . Then

$$ \begin{align*} m\square a=(m\square b_0)\square a=m\square(b_0a)=\varphi(b_0a)m=\varphi(a)m \end{align*} $$

for all $a\in {\mathcal A}$ . This shows that m is a right invariant $\varphi $ -mean on ${\mathcal A}^*$ .

Before giving the next result, we recall that a Banach algebra ${\mathcal A}$ is weakly sequentially complete if every weakly Cauchy sequence in ${\mathcal A}$ is weakly convergent in ${\mathcal A}$ . For example, preduals of von Neumann algebras are weakly sequentially complete (see [Reference Takesaki15]).

Proof By assumption and Theorem 3.2, we conclude that $\operatorname {WAP}(I)$ has a right invariant $1$ -mean. Since I is Arens regular, we have that $\operatorname {WAP}(I)=I^*$ . This implies that I is right $1$ -amenable. Now, by Proposition 3.5, we obtain that $L^1({\mathbb G})$ is right $1$ -amenable or equivalently, ${\mathbb G}$ is amenable. Thus, there is an invariant $1$ -mean on $L^{\infty }({\mathbb G})$ . Again by two-sided version of Proposition 3.5, we conclude that there is an invariant $1$ -mean m on $I^*$ . Since I is Arens regular and weakly sequentially complete, it follows from [Reference Kaniuth, Lau and Pym7, Theorem 3.9] that $m\in I$ . Therefore, for every $f\in L^1({\mathbb G})$ , we have

$$ \begin{align*} f*m=f*(m*m)=(f*m)*m=\langle f*m, 1\rangle m=\langle f, 1\rangle m. \end{align*} $$

Thus, m is a left invariant $1$ -mean belonging to $L^1({\mathbb G})$ , and equivalently ${\mathbb G}$ is compact (see [Reference Bédos and Tuset1, Proposition 3.1]).

Proof If I is reflexive, then I is clearly Arens regular. Conversely, suppose that I is Arens regular. Then ${\mathbb G}$ is compact by Proposition 3.6 and so by [Reference Runde13, Theorem 3.8], $L^1({\mathbb G})$ is an ideal in its bidual. Since I has a bounded approximate identity, Cohen’s Factorization theorem implies that $I*I=\{a*b: a, b\in I\}=I$ . Hence, we drive that

$$ \begin{align*} I\square I^{**}=(I*I)\square I^{**}\subseteq I\square(I\square L^1({\mathbb G})^{**})\subseteq I*L^1({\mathbb G})\subseteq I. \end{align*} $$

This shows that I is a right ideal in its bidual. Thus, by [Reference Ulger16, Corollar ies 3.7 and 3.9], we obtain that I is reflexive.

Dually to [Reference Forrest5, Proposition 3.14], we obtain the result below for the group algebra $L^1(G)$ of a locally compact group G. We would like to recall that $\operatorname {WAP}(L^1(G))$ admits an invariant mean.

4 Convolution trace class operators

We recall from [Reference Lau10] that a Lau algebra ${\mathcal A}$ is a Banach algebra such that ${\mathcal A}^*$ is a von Neumann algebra whose unit $1$ lies in the spectrum of ${\mathcal A}$ . Let ${\mathbb G}$ be a locally compact quantum group. Then it is easy to see that $1=1\circ \pi \in \operatorname {sp}(\mathcal {T}_{\triangleright }({\mathbb G}))$ . Now, since $B(L^{2}(\mathbb {G}))$ is a von Neumann algebra, it follows that $\mathcal {T}_{\triangleright }({\mathbb G})$ is a Lau algebra. In this section, we are interested to study the relation between the existence of left or right invariant $1$ -means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ and on $\operatorname {WAP}(L^1(\mathbb {G}))$ .

Lemma 4.1 Let ${\mathbb G}$ be a locally compact quantum group. Then

$$ \begin{align*} \operatorname{WAP}(\mathcal{T}_{\triangleright}({\mathbb G}))\triangleright \mathcal{T}_{\triangleright}({\mathbb G})\subseteq \operatorname{WAP}(L^{1}(\mathbb{G})). \end{align*} $$

Proof Suppose that $x\in \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ and $w\in \mathcal {T}_{\triangleright }({\mathbb G})$ . Let $(f_k)_k$ be a bounded sequence in $L^1(\mathbb {G})$ . For each k, let $w_k\in \mathcal {T}_{\triangleright }({\mathbb G})$ be a normal extension of $f_k$ . By weak compactness of the map $\lambda _x: \mathcal {T}_{\triangleright }({\mathbb G})\rightarrow B(L^{2}(\mathbb {G}))$ , there is a subsequence $(w_{k_j})$ of $(w_k)$ such that $(w_{k_j}\triangleright x)$ converges weakly in $B(L^{2}(\mathbb {G}))$ to some $y\in B(L^{2}(\mathbb {G}))$ . It is easy to check that $(w_{k_j}\triangleright x \triangleright w)$ converges weakly in $B(L^{2}(\mathbb {G}))$ to $y \triangleright w$ . Now, let $m\in L^{\infty }(\mathbb {G})^*$ , and let $\widetilde {m}\in B(L^{2}(\mathbb {G}))^*$ be a Hahn–Banach extension of m. Since $B(L^{2}(\mathbb {G}))\triangleright \mathcal {T}_{\triangleright }({\mathbb G})\subseteq L^{\infty }(\mathbb {G})$ , we have

$$ \begin{align*} \langle m, f_{k_j}\cdot(x\triangleright w)\rangle= \langle \widetilde{m}, w_{k_j} \triangleright x\triangleright w \rangle\rightarrow \langle \widetilde{m}, y\triangleright w\rangle= \langle m, y\triangleright w\rangle. \end{align*} $$

This shows that $x\triangleright w\in \operatorname {WAP}(L^{1}(\mathbb {G}))$ .

Proof Let m be a right invariant $1$ -mean on $\operatorname {WAP}(L^{1}(\mathbb {G}))$ . Define $\widetilde {m}\in \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))^*$ by $\langle \widetilde {m}, x\rangle =\langle m, x\triangleright w_0\rangle $ for all $x\in \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ , where $w_0\in \mathcal {T}_{\triangleright }({\mathbb G})$ with $\|w_0\|=\langle w_0 ,1\rangle =1$ . Then it is easy to check that $\langle \widetilde {m}, 1\rangle =1$ . Moreover, we have

$$ \begin{align*} \langle\widetilde{m}, w\triangleright x\rangle&=\langle m, w\triangleright (x\triangleright w_0)\rangle\\ &=\langle m, \pi(w)\cdot(x\triangleright w_0)\rangle\\ &= \langle w, 1\rangle\langle m, x\triangleright w_0\rangle\\ &=\langle w, 1\rangle\langle\widetilde{m}, x\rangle \end{align*} $$

for all $w\in \mathcal {T}_{\triangleright }({\mathbb G})$ and $x\in \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ , proving that $\widetilde {m}$ is a right invariant $1$ -mean on $\operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ .

Conversely, suppose that n is a right invariant $1$ -mean on $\operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ . Since $\pi : \mathcal {T}_{\triangleright }({\mathbb G})\rightarrow L^{1}(\mathbb {G})$ is a continuous algebra homomorphism, it follows from [Reference Young18, Corollary to Lemma 1] that the map $\pi ^*$ maps $\operatorname {WAP}(L^{1}(\mathbb {G}))$ to $\operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ . Thus, we can define $\widetilde {n}\in \operatorname {WAP}(L^{1}(\mathbb {G}))^*$ by $\widetilde {n}:=n\circ \pi ^*$ . It is easily verified that $\langle \widetilde {n}, 1\rangle =1$ . For every $f\in L^1(\mathbb {G})$ and $x\in \operatorname {WAP}(L^{1}(\mathbb {G}))$ , let $w\in \mathcal {T}_{\triangleright }({\mathbb G})$ be a normal extension of f. Then we have

$$ \begin{align*} \langle\pi^*(f\cdot x),w'\rangle=\langle f\cdot x,\pi(w')\rangle &=\langle x, \pi(w')*\pi(w)\rangle\\ &=\langle\pi^*(x),w'\triangleright w\rangle\\ &=\langle w\triangleright \pi^*(x),w'\rangle, \end{align*} $$

for all $w'\in \mathcal {T}_{\triangleright }({\mathbb G})$ . Therefore,

$$ \begin{align*} \langle\widetilde{n}, f\cdot x\rangle&=\langle n, \pi^*(f\cdot x)\rangle\\ &=\langle n, w\triangleright \pi^*(x)\rangle\\ &=\langle w, 1\rangle\langle n, \pi^*(x)\rangle\\ &=\langle f, 1\rangle\langle\widetilde{n}, x\rangle. \end{align*} $$

That is, $\widetilde {n}$ is a right invariant $1$ -mean on $\operatorname {WAP}(L^{1}(\mathbb {G}))$ .

Before giving the next result, recall that if ${\mathbb G}= L^{\infty }(G)$ for a locally compact group G, then $\mathcal {T}_{\triangleright }(\mathbb {G})$ is the convolution algebra introduced by Neufang in [Reference Neufang11].

Proof Let m be a left invariant $1$ -mean on $\operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ . Then for every $x\in \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ , we have $m\cdot x=\langle m, x\rangle 1$ , by left invariance. Now, consider the map

$$ \begin{align*}E:\operatorname{WAP}(\mathcal{T}_{\triangleright}({\mathbb G}))\rightarrow \operatorname{WAP}(\mathcal{T}_{\triangleright}({\mathbb G}))\end{align*} $$

defined by $E(x)=m\cdot x=\langle m, x\rangle 1$ for all $x\in \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ . Then for every $\hat {x}\in L^{\infty }(\hat {\mathbb {G}})$ , we have

$$ \begin{align*}E(\hat{x})=m\cdot \hat{x}=\langle m, 1\rangle\hat{x}=\hat{x}.\end{align*} $$

These prove that $L^{\infty }(\widehat {\mathbb {G}})=E(L^{\infty }(\widehat {\mathbb {G}}))\subseteq \mathbb {C}1$ . Therefore, $L^{\infty }(\widehat {\mathbb {G}})=\mathbb {C}1$ and so ${\mathbb G}$ is trivial.