Multiplier norm of finite subsets of discrete groups

Abstract

Let G G be a discrete group, V N ( G ) VN(G) the von Neumann algebra generated by the left translation operators and A ( G ) A(G) the Fourier algebra of G G . Then V N ( G ) VN(G) can be considered as a subspace of ℓ 2 ( G ) \ell ^2(G) and A ( G ) = ℓ 2 ( G ) ∗ ℓ 2 ( G ) A(G) = \ell ^2(G)*\ell ^2(G) . The Banach space dual of A ( G ) A(G) is V N ( G ) VN(G) . The duality between φ ∈ V N ( G ) \varphi \in VN(G) and u ∈ A ( G ) u \in A(G) is ⟨ φ , u ⟩ = ∑ x ∈ G φ ( x ) u ( x ) \langle \varphi ,u \rangle = \sum _{x \in G}{\varphi (x)u(x)} , if φ ∈ ℓ 1 ( G ) \varphi \in \ell ^{1}(G) or u ∈ ℓ 2 ( G ) u \in \ell ^2(G) ; in these two cases, φ u ∈ ℓ 1 ( G ) \varphi u \in \ell ^{1}(G) . We will show that if G G is infinite then there exist φ ∈ V N ( G ) \varphi \in VN(G) and u ∈ A ( G ) u \in A(G) such that φ u ∉ ℓ 1 ( G ) \varphi u \notin \ell ^{1}(G) . When G G is countably infinite, this implies that there exist φ ∈ V N ( G ) \varphi \in VN(G) , u ∈ A ( G ) u \in A(G) and an increasing sequence of finite subsets F n F_n , G = ⋃ F n G = \bigcup F_n such that the sequence ∑ x ∈ F n φ ( x ) u ( x ) \sum _{x \in F_n}{\varphi (x)u(x)} is divergent and therefore ‖ χ F n ‖ M A ( G ) \|{\chi _{F}}_n \|_{MA(G)} is unbounded. This leads us to give a preliminary study of the growth of ‖ χ F ‖ M A ( G ) \|\chi _{F}\|_{MA(G)} as | F | |F| , the size of the finite subset F F of G G , is increasing. Recall that when G = Z G = \mathbb {Z} , the additive group of integers, ‖ χ F ‖ M A ( Z ) = 1 2 π ∫ 0 2 π | ∑ k ∈ F e i k x | d x \|\chi _{F}\|_{MA(\mathbb {Z})} = \frac {1}{2\pi }\int _0^{2\pi } \big |\sum _{k \in F} {e^{ikx}}\big | \, d{x} .