Amongst other results, we show that if $\mathcal{A}(G)$ is either $AM(G)$ or $A_{cb}(G)$, then $UCB(\mathcal{A}(G)) \subseteq \mathcal{A}P(A(G))$ if and only if $G$ is discrete. We also show that if $UCB(\mathcal{A}(G)) = \mathcal{A}(G)^\ast$, then every amenable closed subgroup of $G$ is compact.

Let $i : A(G) \to \mathcal{A}(G)$ be the natural injection. We show that if $X$ is any closed topologically introverted subspace of $\mathcal{A}(G)^\ast$ that contains $L^1(G)$, then $i^\ast (X)$ is closed in $A(G)$ if and only if $G$ is amenable.

PDF XML]]>Amongst other results, we show that if $\mathcal{A}(G)$ is either $AM(G)$ or $A_{cb}(G)$, then $UCB(\mathcal{A}(G)) \subseteq \mathcal{A}P(A(G))$ if and only if $G$ is discrete. We also show that if $UCB(\mathcal{A}(G)) = \mathcal{A}(G)^\ast$, then every amenable closed subgroup of $G$ is compact.

Let $i : A(G) \to \mathcal{A}(G)$ be the natural injection. We show that if $X$ is any closed topologically introverted subspace of $\mathcal{A}(G)^\ast$ that contains $L^1(G)$, then $i^\ast (X)$ is closed in $A(G)$ if and only if $G$ is amenable.

PDF XML]]>Let $V$ be open connected and $U$ be open, or both $U,V$ are convex bodies, in normed linear spaces $E,F$, respectively. Let $Tf = h \cdot f \circ \varphi$ be a bijective weighed composition operator between the vector spaces $\Lip(U)$ and $\Lip(V)$, $\Lipb(U)$ and $\Lipb(V),$ $\Liploc(U)$ and $\Liploc(V)$, or $\Lippt(U)$ and $\Lippt(V)$, preserving the Lipschitz, locally Lipschitz, or pointwise Lipschitz constants, respectively.We show that there is a linear isometry $A : F \to E$, an $\alpha \gt 0$ and a vector $b \in E$ such that $\varphi(x) = \alpha Ax+b$, and $h$ is a constant function assuming value $\pm 1 / \alpha$. More concrete results are obtained for the special cases when $E = F = \mathbb{R}^n$, or when $U, V$ are $n$-dimensional flat manifolds.

PDF XML]]>Let $V$ be open connected and $U$ be open, or both $U,V$ are convex bodies, in normed linear spaces $E,F$, respectively. Let $Tf = h \cdot f \circ \varphi$ be a bijective weighed composition operator between the vector spaces $\Lip(U)$ and $\Lip(V)$, $\Lipb(U)$ and $\Lipb(V),$ $\Liploc(U)$ and $\Liploc(V)$, or $\Lippt(U)$ and $\Lippt(V)$, preserving the Lipschitz, locally Lipschitz, or pointwise Lipschitz constants, respectively.We show that there is a linear isometry $A : F \to E$, an $\alpha \gt 0$ and a vector $b \in E$ such that $\varphi(x) = \alpha Ax+b$, and $h$ is a constant function assuming value $\pm 1 / \alpha$. More concrete results are obtained for the special cases when $E = F = \mathbb{R}^n$, or when $U, V$ are $n$-dimensional flat manifolds.

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