1 Introduction
Let ${\mathcal A}$ be a Banach algebra. Then ${\mathcal A}^*$ is canonically a Banach ${\mathcal A}$ -bimodule with the actions
for all $a,b\in {\mathcal A}$ and $x\in {\mathcal A}^*$ . There are two naturally defined products, which we denote by $\square $ and $\Diamond $ on the second dual $\mathcal {A}^{* *}$ of $\mathcal {A},$ each extending the product on $\mathcal {A}$ . For $m, n \in \mathcal {A}^{* *}$ and $x \in \mathcal {A}^{*},$ the first Arens product $\square $ in $\mathcal {A}^{* *}$ is given as follows:
where $n \cdot x \in \mathcal {A}^{*}$ is defined by $\langle n \cdot x, a\rangle =\langle n, x \cdot a\rangle $ for all $a \in \mathcal {A}$ . Similarly, the second Arens product $\Diamond $ in $\mathcal {A}^{* *}$ satisfies
where $x \cdot m \in \mathcal {A}^{*}$ is given by $\langle x \cdot m, a\rangle =\langle m, a \cdot x\rangle $ for all $a \in \mathcal {A} .$ The Banach algebra $\mathcal {A}$ is called Arens regular if $\square $ and $\Diamond $ coincide on $\mathcal {A}^{* *}$ .
We denote the spectrum of $\mathcal {A}$ by $\operatorname {sp}(\mathcal {A})$ . Let $\varphi \in \operatorname {sp}(\mathcal {A}),$ and let X be a Banach right $\mathcal {A}$ -submodule of ${\mathcal A}^*$ with $\varphi \in X$ . Then a left invariant $\varphi $ -mean on X is a functional $m\in X^*$ satisfying
Right and (two-sided) invariant $\varphi $ -means are defined similarly. The Banach algebra $\mathcal {A}$ is called left $\varphi $ -amenable if there exists a left invariant $\varphi $ -mean on ${\mathcal A}^*$ (see [Reference Kaniuth, Lau and Pym7]). This notion generalizes the concept of left amenability for Lau algebras, a class of Banach algebras including all convolution quantum group algebras, which was first introduced and studied in [Reference Lau10].
A Banach right (resp. left) $\mathcal {A}$ -submodule X of $\mathcal {A}^{*}$ is called left (resp. right) introverted if $X^{*} \cdot X \subseteq X$ (resp. $X \cdot X^* \subseteq X$ ). In this case, $X^{*}$ is a Banach algebra with the multiplication induced by the first (resp. second) Arens product $\square $ (resp. $\Diamond $ ) inherited from $\mathcal {A}^{* *}$ . A Banach $\mathcal {A}$ -subbimodule X of $\mathcal {A}^{*}$ is called introverted if it is both left and right introverted (see [Reference Dales and Lau2, Chapter 5] for details).
An element x of $\mathcal {A}^{*}$ is weakly almost periodic if the map $\lambda _x: a \mapsto a\cdot x $ from $\mathcal {A}$ into $\mathcal {A}^{*}$ is a weakly compact operator. Let $\operatorname {WAP}(\mathcal {A})$ denote the closed subspace of ${\mathcal A}^*$ consisting of the weakly almost periodic functionals on $\mathcal {A}$ . Then $\operatorname {WAP}(\mathcal {A})$ is an introverted subspace of $\mathcal {A}^{*}$ containing $\operatorname {sp}(\mathcal {A})$ . We would like to recall from [Reference Dales and Lau2, Proposition 3.11] that $m\square n= m\Diamond n $ for all $m,n\in \operatorname {WAP}(\mathcal {A})^*$ . Now suppose that I is a closed ideal in ${\mathcal A}$ with a bounded approximate identity. Then, by [Reference Dales and Lau2, Proposition 3.12] $\operatorname {WAP}(I)$ is a neo-unital Banach I-bimodule; that is, $\operatorname {WAP}(I) = I\cdot \operatorname {WAP}(I) = \operatorname {WAP}(I)\cdot I$ . Moreover, $\operatorname {WAP}(I)$ becomes a Banach ${\mathcal A}$ -bimodule (see [Reference Runde14, Proposition 2.1.6]).
In the case that A is the group algebra $L^1(G)$ of a locally compact group G, it is known that $\operatorname {WAP}(L^1(G))$ admits an invariant mean, which is unique, that is, a norm one functional $m\in L^1(G)^{**}$ with $\langle m, 1\rangle = 1$ and
for all $x\in \operatorname {WAP}(L^1(G))$ and $f\in L^1(G)$ (see [Reference Wong17]).
Furthermore, it is known from [Reference Dales, Lau and Strauss3, Proposition 5.16] that if G is discrete or amenable, then $\operatorname {WAP}(M(G))$ admits an invariant mean, which is unique, where $M(G)$ denotes the measure algebra of G. Recently, Neufang in [Reference Neufang12] generalized this latter result to arbitrary locally compact groups, thereby answering a question posed in [Reference Dales, Lau and Strauss3].
In this article, we generalize the main result of [Reference Neufang12] to an arbitrary Banach algebra ${\mathcal A}$ . More precisely, for $\varphi \in \operatorname {sp}({\mathcal A})$ , we show that if I is a closed ideal of ${\mathcal A}$ with a bounded approximate identity such that $I\not \subseteq \ker \varphi $ , then $\operatorname {WAP}(\mathcal {A})$ admits a right (left) invariant $\varphi $ -mean if and only if $\operatorname {WAP}(I)$ admits a right (left) invariant $\varphi |_I$ -mean. Applying our results to algebras over locally compact (quantum) groups, we show that, if I is a closed ideal of $L^1(G)$ with a bounded approximate identity such that $I\not \subseteq \ker 1$ , then I is Arens regular if and only if it is reflexive.
Finally, for a locally compact quantum group ${\mathbb G}$ , we characterize the existence of left and right invariant $1$ -means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ , where $\mathcal {T}_{\triangleright }({\mathbb G})$ denotes the trace class operators on $L^2({\mathbb G})$ , but equipped with a product different from composition (see [Reference Hu, Neufang and Ruan6].
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
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
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
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
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
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
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
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
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
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
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
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:
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
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
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:
It follows that
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.
Theorem 3.2 Let ${\mathcal A}$ be a Banach algebra with $\varphi \in \operatorname {sp}(\mathcal {A})$ , and let I be a closed ideal of ${\mathcal A}$ with a bounded approximate identity such that $I\not \subseteq \ker \varphi $ . Then the following statements are equivalent:
-
(i) $\mathrm {WAP}(I)$ has a right (left) invariant $\varphi |_I$ -mean.
-
(ii) $\mathrm {WAP}(A)$ has a right (left) invariant $\varphi $ -mean.
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
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
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:
It is easily verified that
Moreover, for every $b\in I$ and $x \in \operatorname {WAP}(I)$ , we have
Therefore, $\tilde {m}$ is a right $\left .\varphi \right |{}_{I}$ -mean on $\operatorname {WAP}(I)$ .
Remark 3.3 We would like to point out the following fact related to right and left invariant $\varphi $ -means on $\mathrm {WAP}({\mathcal A})$ . Suppose that m is a left invariant $\varphi $ -mean and n is a right invariant $\varphi $ -mean on $\mathrm {WAP}({\mathcal A})$ . Using weak $^*$ -continuity of the maps $p\mapsto p\square m$ and $p\mapsto n\Diamond p$ on $\mathrm {WAP}({\mathcal A})^*$ , we obtain that $m=n(\varphi )m=n\square m=n\Diamond m=m(\varphi )n=n$ . In particular, if there is an invariant $\varphi $ -mean on $\mathrm {WAP}({\mathcal A})$ , then it is unique.
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
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.
Corollary 3.4 Let ${\mathbb G}$ be a co-amenable locally compact quantum group. Then $\operatorname {WAP}(L^1({\mathbb G}))$ has a right invariant $1$ -mean or equivalently has an invariant $1$ -mean if and only if $\operatorname {WAP}(M({\mathbb G}))$ has an invariant $1$ -mean.
Proposition 3.5 Let ${\mathcal A}$ is a Banach algebra, and let I is a closed ideal in ${\mathcal A}$ . Let $\varphi \in \operatorname {sp}(\mathcal {A})$ be such that $I\not \subseteq \ker \varphi $ . Then ${\mathcal A}^*$ admits a right invariant $\varphi $ -mean if and only if $I^*$ admits a right invariant $\varphi |_I$ -mean.
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
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
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]).
Proposition 3.6 Let ${\mathbb G}$ be a locally compact quantum group such that $\operatorname {WAP}(L^1({\mathbb G}))$ has an invariant $1$ -mean, and let I be a closed ideal of $L^1({\mathbb G})$ with a bounded approximate identity such that $I\not \subseteq \ker 1$ . If I is Arens regular, then ${\mathbb G}$ is compact.
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
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]).
Theorem 3.7 Let ${\mathbb G}$ be a locally compact quantum group such that $\operatorname {WAP}(L^1({\mathbb G}))$ has an invariant $1$ -mean, and let I be a closed ideal of $L^1({\mathbb G})$ with a bounded approximate identity such that $I\not \subseteq \ker 1$ . Then I is Arens regular if and only if it is reflexive.
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
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.
Corollary 3.8 Let G be a locally compact group, and let I be a closed ideal of $L^1(G)$ with a bounded approximate identity such that $I\not \subseteq \ker 1$ . Then I is Arens regular if and only if it is reflexive.
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
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
This shows that $x\triangleright w\in \operatorname {WAP}(L^{1}(\mathbb {G}))$ .
Theorem 4.2 Let ${\mathbb G}$ be a locally compact quantum group. Then $\operatorname {WAP}(L^{1}(\mathbb {G}))$ has a right invariant $1$ -mean if and only if $\operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ has a right invariant $1$ -mean.
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
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
for all $w'\in \mathcal {T}_{\triangleright }({\mathbb G})$ . Therefore,
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].
Corollary 4.3 Let G be a locally compact group, and let ${\mathbb G}= L^{\infty }(G)$ . Then $\operatorname {WAP}(\mathcal {T}_{\triangleright }(\mathbb {G}))$ admits a right invariant $1$ -mean.
Theorem 4.4 Let $\mathbb {G}$ be a locally compact quantum group. Then $\operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ has a left invariant $1$ -mean if and only if $\mathbb {G}$ is trivial.
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
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
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.
Acknowledgment
The authors are grateful to the referee for his/her careful reading of the paper and valuable suggestions.