Amenability, weak amenability and bounded approximate identities in multipliers of the Fourier algebra
Tom 273 / 2023
Streszczenie
We focus on problems concerning amenability and weak amenability in the algebras $A_{\rm cb}(G)$ and $A_M(G)$ which are the closures of the Fourier algebras $A(G)$ in the spaces of completely bounded multipliers of $A(G)$ and the multipliers of $A(G)$ respectively. We show that either algebra is weakly amenable if and only if the connected component $G_e$ of $G$ is abelian. We also show that if either algebra is amenable, then $G$ has an open amenable subgroup. Moreover, if $G$ is almost connected, then either algebra is amenable if and only if $G$ is virtually abelian.
Let $ \mathcal A(G)$ be either $A_{\rm cb}(G)$ or $A_M(G)$. Assume that $\mathcal A(G)$ has a bounded approximate identity. If $G$ is a [SIN]-group and $E$ is an element of $ \mathcal R(G)$, the closed coset ring of $G$, then the ideal $I_{\mathcal A(G)}(E)$ consisting of all functions in $\mathcal A(G)$ that vanish on $E$ also has a bounded approximate identity. In particular, if $\mathcal A(G)$ is (weakly) amenable, then so is $I_{\mathcal A(G)}(E)$.
