Consequence of Hahn-Banach - norm closed convex hull contains $0$

I am reading Davidson's '$C^*$ algebras by example'. I am currently in the section about group $C^*$ algebras, in the proof that for an amenable discrete Hausdorff group $G$, the reduced group $C^*$ algebra is isomorphic to the groups $C^*$ algebra (theorem $VII.2.8$). I am having trouble with understanding a certain part of the proof, which uses Hahn-Banach in a way I am not familiar with.

Before this part, he chooses a left invariant mean $m\in\ell^{\infty}(G)^*$ (which exists due to amenability). He then approximates it using a net in $\ell^1$ - $(g_\gamma)_{\gamma\in \Gamma}$. He then shows that in the weak star topology: $$\lim_\gamma (\delta_s*g_\gamma-g_\gamma)=0$$ Here, $*$ is the convolution in $\ell^1(G)$, and $\delta_s$ is the sequence with $1$ at the $s$ entry ($s$ is some element in $G$) and $0$ everywhere else.

Now he says the following:

By the Hahn-Banach theorem, for any finite subset $s_1,...,s_n\in G$, the norm closed convex hull of the set: $$\{(\delta_{s_i}*g_\gamma-g_\gamma)_{1\leq i \leq n}:\gamma \in \Gamma\}\subset \ell^1(G)^n$$ Contains $0$.

The problem is, I have no idea how this statement follows from Hahn-Banach. Does anyone here have an idea?

Thanks in advance.

Solution 1:

I will sketch how you can (probably) do this (I didn't check any details myself, so take this answer with a grain of salt). I can add more details if necessary.

(STEP 1) Show that $0$ is in the weak closure of the convex hull of the given set. Most likely it will be useful to use the duality $(\ell^1)^* = \ell^\infty$ for this.

(STEP 2) Use the Hahn-Banach separation theorem to prove the following general functional analytic statement:

If $C$ is a convex subset of a locally convex Hausdorff topological vector space $(V, \tau)$ and if $\tau_w$ is the weak topology on $V$ (w.r.t. $\tau$), then $$\overline{C}^\tau = \overline{C}^{\tau_w}.$$

Sketch: One inclusion is obvious. For the other one, assume to the contrary that the inclusion is strict and use the Hahn-Banach separation theorem to separate a point and a closed set.

(STEP 3) Combine the first two steps to conclude that $0$ is in the norm-closure of the convex hull of the given set.