Multiplier norm of finite subsets of discrete groups

Abstract:

Let $G$ be a discrete group, $VN(G)$ the von Neumann algebra generated by the left translation operators and $A(G)$ the Fourier algebra of $G$. Then $VN(G)$ can be considered as a subspace of $\ell ^2(G)$ and $A(G) = \ell ^2(G)*\ell ^2(G)$. The Banach space dual of $A(G)$ is $VN(G)$. The duality between $\varphi \in VN(G)$ and $u \in A(G)$ is $\langle \varphi ,u \rangle = \sum _{x \in G}{\varphi (x)u(x)}$, if $\varphi \in \ell ^{1}(G)$ or $u \in \ell ^2(G)$; in these two cases, $\varphi u \in \ell ^{1}(G)$. We will show that if $G$ is infinite then there exist $\varphi \in VN(G)$ and $u \in A(G)$ such that $\varphi u \notin \ell ^{1}(G)$. When $G$ is countably infinite, this implies that there exist $\varphi \in VN(G)$, $u \in A(G)$ and an increasing sequence of finite subsets $F_n$, $G = \bigcup F_n$ such that the sequence $\sum _{x \in F_n}{\varphi (x)u(x)}$ is divergent and therefore $\|{\chi _{F}}_n \|_{MA(G)}$ is unbounded. This leads us to give a preliminary study of the growth of $\|\chi _{F}\|_{MA(G)}$ as $|F|$, the size of the finite subset $F$ of $G$, is increasing. Recall that when $G = \mathbb {Z}$, the additive group of integers, $\|\chi _{F}\|_{MA(\mathbb {Z})} = \frac {1}{2\pi }\int _0^{2\pi } \big |\sum _{k \in F} {e^{ikx}}\big | \, d{x}$.