A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].

Let $G$ be a compact Abelian group and $E$ a subset of the group $\widehat {G}$ of continuous characters of $G$. We study Arens regularity-related properties of the ideals $L_E^1(G)$ of $L^1(G)$ that are made of functions whose Fourier transform is supported on $E\subseteq \widehat {G}$. Arens regularity of $L_E^1(G)$, the centre of $L_E^1(G)^{\ast \ast }$ and the size of $L_E^1(G)^\ast /\mathcal {WAP}(L_E^1(G))$ are studied. We establish general conditions for the regularity of $L_E^1(G)$ and deduce from them that $L_E^1(G)$ is not strongly Arens irregular if $E$ is a small-2 set (i.e. $\mu \ast \mu \in L^1(G)$ for every $\mu \in M_E^1(G)$), which is not a $\Lambda (1)$-set, and it is extremely non-Arens regular if $E$ is not a small-2 set. We deduce also that $L_E^1(G)$ is not Arens regular when $\widehat {G}\setminus E$ is a Lust-Piquard set.

Let ${\mathcal A}$ be a Banach algebra, and let $\varphi $ be a nonzero character on ${\mathcal A}$. For a closed ideal I of ${\mathcal A}$ with $I\not \subseteq \ker \varphi $ such that I has a bounded approximate identity, we show that $\operatorname {WAP}(\mathcal {A})$, the space of weakly almost periodic functionals on ${\mathcal A}$, admits a right (left) invariant $\varphi $-mean if and only if $\operatorname {WAP}(I)$ admits a right (left) invariant $\varphi |_I$-mean. This generalizes a result due to Neufang for the group algebra $L^1(G)$ as an ideal in the measure algebra $M(G)$, for a locally compact group G. Then we apply this result to the quantum group algebra $L^1({\mathbb G})$ of a locally compact quantum group ${\mathbb G}$. Finally, we study the existence of left and right invariant $1$-means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$.

We prove that for a Banach algebra A having a bounded $\mathcal {Z}(A)$ -approximate identity and for every $\mathbf {[IN]}$ group G with a weight w which is either constant on conjugacy classes or satisfies $w \geq 1$ , $\mathcal {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) \otimes ^{\gamma } \mathcal {Z}(A)$ . As an application, we discuss the conditions under which $\mathcal {Z}(L^{1}_{\omega }(G,A))$ enjoys certain Banach algebraic properties, such as weak amenability or semisimplicity.

Let $\mathbb{L}$ be a length function on a group $G$, and let ${{M}_{\mathbb{L}}}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on ${{\ell }^{2}}\left( G \right)$. Following Connes, ${{M}_{\mathbb{L}}}$ can be used as a “Dirac” operator for the reduced group ${{C}^{*}}$-algebra $C_{r}^{*}\left( G \right)$. It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$, which defines a metric on the state space of $C_{r}^{*}\left( G \right)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Let $\text{H}$ be a subgroup of some locally compact group $\text{G}$. Assume that $\text{H}$ is approximable by discrete subgroups and that $\text{G}$ admits neighborhood bases which are almost invariant under conjugation by finite subsets of $\text{H}$. Let $m:\text{G}\rightarrow \mathbb{C}$ be a bounded continuous symbol giving rise to an $L_{p}$-bounded Fourier multiplier (not necessarily completely bounded) on the group von Neumann algebra of $\text{G}$ for some $1\leqslant p\leqslant \infty$. Then, $m_{\mid _{\text{H}}}$ yields an $L_{p}$-bounded Fourier multiplier on the group von Neumann algebra of $\text{H}$ provided that the modular function ${\rm\Delta}_{\text{G}}$ is equal to 1 over $\text{H}$. This is a noncommutative form of de Leeuw’s restriction theorem for a large class of pairs $(\text{G},\text{H})$. Our assumptions on $\text{H}$ are quite natural, and they recover the classical result. The main difference with de Leeuw’s original proof is that we replace dilations of Gaussians by other approximations of the identity for which certain new estimates on almost-multiplicative maps are crucial. Compactification via lattice approximation and periodization theorems are also investigated.

For a Lie group $G$, we show that the map $C_{c}^{\infty }\,\left( G \right)\,\times \,C_{c}^{\infty }\,\left( G \right)\,\to \,C_{c}^{\infty }\,\left( G \right),\,\left( \gamma ,\,\eta \right)\mapsto \,\gamma \,*\,\eta $ taking a pair of test functions to their convolution, is continuous if and only if $G$ is $\sigma -$compact. More generally, consider $r,\,s,\,t\,\in {{\mathbb{N}}_{0}}\,\cup \,\left\{ \infty \right\}$ with $t\,\le \,r\,+\,s$, locally convex spaces ${{E}_{1}}\,,\,{{E}_{2}}$ and a continuous bilinear map $b:\,{{E}_{1}}\,\times \,{{E}_{2}}\,\to \,F$ to a complete locally convex space $F$. Let $\beta :\,C_{c}^{r}\,\left( G,\,{{E}_{1}} \right)\,\times \,C_{c}^{S}\,\left( G,\,{{E}_{2}} \right)\,\to$$C_{c}^{t}\,\left( G,\,F \right),\,\left( \gamma ,\,\eta \right)\,\mapsto \,\gamma \,*\,b\,\eta$ be the associated convolution map. The main result is a characterization of those $\left( G,\,r,s,t,b \right)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported ${{L}^{1}}$-functions and convolution of compactly supported Radon measures.

We study locally compact quantum groups $\mathbb{G}$ through the convolution algebras ${{L}_{1}}\left( \mathbb{G} \right)$ and $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\triangleright \right)$. We prove that the reduced quantum group ${{C}^{*}}$ -algebra ${{C}_{0}}\left( \mathbb{G} \right)$ can be recovered from the convolution $\triangleright $ by showing that the right $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$-module $\left\langle K\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,T\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ is equal to ${{C}_{0}}\left( \mathbb{G} \right)$. On the other hand, we show that the left $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$-module $\left\langle T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,K\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ is isomorphic to the reduced crossed product ${{C}_{0}}\left( \widehat{\mathbb{G}} \right){{\,}_{r}}\,\ltimes \,{{C}_{0}}\left( \mathbb{G} \right)$, and hence is a much larger ${{C}^{*}}$ -subalgebra of $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$.

We establish a natural isomorphism between the completely bounded right multiplier algebras of ${{L}_{1}}\left( \mathbb{G} \right)$ and $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\,\triangleright \right)$, and settle two invariance problems associated with the representation theorem of Junge–Neufang–Ruan (2009). We characterize regularity and discreteness of the quantum group $\mathbb{G}$ in terms of continuity properties of the convolution $\triangleright $ on $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$. We prove that if $\mathbb{G}$ is semiregular, then the space $\left\langle T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,B\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ of right $\mathbb{G}$-continuous operators on ${{L}_{2}}\left( \mathbb{G} \right)$, which was introduced by Bekka (1990) for ${{L}_{\infty }}\left( G \right)$, is a unital ${{C}^{*}}$-subalgebra of $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$. In the representation framework formulated by Neufang–Ruan–Spronk (2008) and Junge–Neufang–Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$. We also characterize some commutation relations of completely bounded multipliers of $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\,\triangleright \right)$ over $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$.

For unimodular semidirect products of locally compact amenable groups $N$ and $H$, we show that one can always construct a Følner net of the form $({{A}_{\alpha }}\,\times \,{{B}_{\beta }})$ for $G$, where $({{A}_{\alpha }})$ is a strong form of Følner net for $N$ and $({{B}_{\beta }})$ is any Følner net for $H$. Applications to the Heisenberg and Euclidean motion groups are provided.

Let $G$ be a compactly generated, locally compact group of polynomial growth. Removing a restrictive technical condition from a previous work, we show that the weighted group algebra $L^{1}_{\omega}(G)$ is a symmetric Banach $*$-algebra if and only if the weight function $\omega $ satisfies the GRS-condition. This condition expresses in a precise technical sense that $\omega $ grows subexponentially.

Let $G$ be a compact group.

Let $\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation

$$\int\limits_{G}{f\left( xty{{t}^{-1}} \right)dt}\,+\int\limits_{G}{f\left( xt\sigma \left( y \right){{t}^{-1}} \right)dt}=2g\left( x \right)h\left( y \right),\,\,\,x,y\in G,$$

where $f,g,h:G\mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d’Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$.

It is proved that the measure algebra M(G) of a locally compact group G is Connes-amenable if and only if G is amenable.

Let VN(G) be the von Neumann algebra of a locally compact group G. We denote by μ the initial ordinal with |μ| equal to the smallest cardinality of an open basis at the unit of G and X = ﹛α ; α < μ﹜.We show that if G is nondiscrete then there exist an isometric *-isomorphism of l∞(X) into VN(G) and a positive linear mapping π of VN(G) onto l∞(X) such that π o = idl∞(X) and and π have certain additional properties. Let UCB((Ĝ)) be the C*–algebra generated by operators in VN(G) with compact support and F(Ĝ) the space of all T∈ VN(G) such that all topologically invariant means on VN(G) attain the same value at T. The construction of the mapping π leads to the conclusion that the quotient space UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) has l∞(X) as a continuous linear image if G is nondiscrete. When G is further assumed to be non-metrizable, it is shown that UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) contains a linear isomorphic copy of l∞(X). Similar results are also obtained for other quotient spaces.