Let $G$ be a weakly amenable group, in the sense that it has a net of finitely supported functions $\varphi:G\to \mathbb{C}$ which converge point wise to 1 and their cb norm is bounded uniformly by some constant $C$. 


It is known that this is equivalent to completely bounded approximation property (CBAP) for the reduced group $C^\*$-algebra of $G$. 


**Question 1:** If $G$ is weakly amenable, does the full group $C^\*$-algebra have CBAP? 


**Question 2:** Do the functions $\varphi$ as multipliers extend to completely bounded maps on the full group $C^\*$-algebra, with the same cb norm? 


**Edit:** Yemon Choi pointed out that the answer to Question 1 is negative. In that case, are there other $C^\*$-completions of the group ring $\mathbb{C}G$ for which the answer(s) would be affirmative?



CBAP for the full group $C^*$-algebra
The answer to the second question is also negative. Let $(u\_g)\_{g\in G}$ be the generating unitaries of $C^{\ast}(G)$. Suppose that $(\varphi\_i)\_{i\in I}$ is a net a functions whose associated multipliers $m\_{\varphi\_{i}}: C^{\ast}\_{r}(G) \to C^{\ast}\_r(G)$ give CBAP of $C^{\ast}\_{r}(G)$. Suppose now that the assignments $u\_g \mapsto \varphi\_i(g) u\_g$ extend to uniformly completely bounded multipliers on the full $C^{\ast}$-algebra $C^{\ast}(G)$. It would follow that $C^{\ast}(G)$ has the CBAP because, by uniform boundedness, it would suffice to check convergence on finite sums $\sum a\_g u\_g$, which are dense in $C^{\ast}(G)$, and this is easy.


Let me now discuss, where the subtlety lies. The multipliers giving CBAP for non-amenable groups are not uniformly bounded in $B(G)$, the Fourier-Stieltjes algebra, and therefore do not give nice multipliers on $C^{\ast}(G)$. Indeed, for a function $\varphi: G \to \mathbb{C}$ with $\|\varphi\|\_{B(G)}\leqslant C$ we get an associated multiplier $m\_{\varphi}:C^{\ast}(G) \to C^{\ast}(G)$ via composition $C^{\ast}(G) \stackrel{\Delta}{\rightarrow} C^{\ast}(G) \otimes C^{\ast}(G) \stackrel{id \otimes \varphi}{\rightarrow} C^{\ast}(G)$, where $\Delta$ is a $\ast$-homomorphism such that $\Delta(u\_g)=u\_g\otimes u\_g$, given by the universal property, and $\varphi:C^{\ast}(G)\to \mathbb{C}$ is the functional given by $\varphi$. Since $\ast$-homomorphisms are completely contractive, we get $\|m\_{\varphi}\|\_{cb}\leqslant C$.


The problematic behaviour of multipliers manifests itself in the fact that they are not *decomposable* as maps from $C^{\ast}\_r(G)$ to itself. What I mean by this is that if we decompose them as combinations of positive multipliers, then the norms of the terms involved are not controlled by the cb norm of the multiplier itself. A positive multiplier on $C^{\ast}\_r(G)$ necessarily comes from a positive definite function on $G$, hence automatically extends to a multiplier on $C^{\ast}(G)$, so I think that decomposability is the real issue here. 


