[Paper Review] Operator amenability of the Fourier algebra in the cb-multiplier norm

Brian Forrest, Volker Runde+1 more|ArXiv.org|Jan 6, 2005

Advanced Operator Algebra Research31 references3 citations

TL;DR

This paper establishes that the Fourier algebra of a discrete, weakly amenable group $G$ equipped with the cb-multiplier norm, denoted $A_{\mathrm{cb}}(G)$, is operator amenable if $G$ is residually finite-dimensional. Notably, $A_{\mathrm{cb}}(\mathbb{F}_2)$ is operator amenable despite $\mathbb{F}_2$ not being amenable, and the authors prove that weakly completely complemented ideals in $A(G)$ correspond precisely to those with cb-bounded approximate identities.

ABSTRACT

Let $G$ be a locally compact group, and let $A_\cb(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_\cb(G)$ is operator amenable. In particular, $A_\cb(F_2)$ is operator amenable even though $F_2$, the free group in two generators, is not an amenable group. Moreover, we show that, if $G$ is a discrete group such that $A_\cb(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the cb-multiplier norm.

Motivation & Objective

• To investigate the operator amenability of $A_{\mathrm{cb}}(G)$, the completion of the Fourier algebra $A(G)$ under the cb-multiplier norm.
• To determine whether $A_{\mathrm{cb}}(G)$ is operator amenable for non-amenable groups, particularly discrete groups like $\mathbb{F}_2$.
• To characterize weakly completely complemented closed ideals in $A(G)$ in terms of approximate identities bounded in the cb-multiplier norm.
• To extend known complementation results for ideals in $A(G)$ beyond the amenable case using operator space techniques.

Proposed method

• Define $A_{\mathrm{cb}}(G)$ as the closure of $A(G)$ in the space of completely bounded multipliers of $A(G)$, equipped with the cb-multiplier norm.
• Use operator space theory and the concept of operator amenability, introduced by Ruan, to analyze $A_{\mathrm{cb}}(G)$ as a quantized Banach algebra.
• Apply module theory over $A_{\mathrm{cb}}(G)$, particularly the notion of completely bounded module projections and duality in $\mathrm{VN}(G)$, to study complementation of ideals.
• Leverage the fact that $A_{\mathrm{cb}}(G)$ is a dual space to extract weak*-limit points of approximate identities, enabling the construction of characteristic functions in $\mathcal{M}_{\mathrm{cb}}(G)$.
• Use the module homomorphism property of projections on $\mathrm{VN}(G)$ and its dual to lift bounded multipliers to $A(G)$, proving that $1_F \in \mathcal{M}_{\mathrm{cb}}(A(G))$ for certain sets $F$.
• Establish equivalence between ideal complementation and existence of cb-bounded approximate identities via duality and module structure.

Experimental results

Research questions

• RQ1 For which non-amenable discrete groups $G$ is $A_{\mathrm{cb}}(G)$ operator amenable?
• RQ2 Can $A_{\mathrm{cb}}(G)$ be operator amenable even when $G$ is not amenable?
• RQ3 What characterizes weakly completely complemented closed ideals in $A(G)$ when $A_{\mathrm{cb}}(G)$ is operator amenable?
• RQ4 Is there a connection between the existence of a cb-bounded approximate identity and the complementation of ideals in $A(G)$?
• RQ5 How do operator space techniques extend classical results on complementation and approximate identities in the Fourier algebra?

Key findings

• For a discrete, weakly amenable group $G$ with $C^*(G)$ residually finite-dimensional, $A_{\mathrm{cb}}(G)$ is operator amenable.
• $A_{\mathrm{cb}}(\mathbb{F}_2)$ is operator amenable, even though $\mathbb{F}_2$ is not amenable, providing the first example of operator amenability for a non-amenable group in this context.
• A closed ideal $I$ of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the $\mathrm{cb}$-multiplier norm.
• If $A_{\mathrm{cb}}(G)$ is operator amenable, then a closed ideal $I$ of $A(G)$ is completely complemented if and only if $I = I(F)$ for some $F \subset G$ with $1_F \in \mathcal{M}_{\mathrm{cb}}(A(G))$.
• The existence of a $\mathrm{cb}$-bounded approximate identity for an ideal $I \subset A(G)$ implies that the characteristic function $1_F$ of the associated set $F$ lies in $\mathcal{M}_{\mathrm{cb}}(A(G))$.
• The proof technique extends results from [Woo1] and [Woo2] to non-amenable groups by using module homomorphisms and duality in $A(G)^{**}$ and $\mathrm{VN}(G)$.