- A C*-algebra $\mathfrak{A}$ is a Banach algebra with an involution operation $* : \left\lbrace\begin{aligned} \mathfrak{A} &\longrightarrow \mathfrak{A} \\ a &\longmapsto a^* \end{aligned} \right.\ $ (antilinear and $\forall\ a,b\in\mathfrak{A},\ (ab)^*=b^*a^*$) such that $ \left\lVert a^*a \right\rVert = \left\lVert a \right\rVert^2\ $ (C* property).
- A state $\omega$ on $\mathfrak{A}$ (assumed unital) is a normalized positive linear functional $\omega: \mathfrak{A}\longrightarrow \mathbb{C}$ i.e. \begin{equation} \forall\ a\in\mathfrak{A},\enspace \omega(a^* a) \geq 0 \ \text{in partic.}\ \omega(a^* a)\in\mathbb{R}\quad \text{ and }\quad \omega(\mathbb{1}_{\mathfrak{A}})= 1 \end{equation}
- Define the left action of $\mathfrak{A}$ on $S(\mathfrak{A})$ the set of all states: $$b\cdot \omega(-) :\equiv \omega(b^*\cdot -\cdot b)$$ and the folium of a state (usually one defines the folium of a representation) as the set of states "generated by the action" of the algebra, i.e. those of the form \begin{equation} \omega_{\mathcal{I}} : \left\lbrace \begin{aligned} \mathfrak{A} &\longrightarrow \quad\mathbb{C} \\ a & \longmapsto \sum_{i\in \mathcal{I}} \omega \left(b^*_i \cdot a \cdot b_i\right) \end{aligned} \right. \qquad\text{with }\ (b_i)_{i\in \mathcal{I}}\in\mathfrak{A}\ \text{ s.t. }\ \sum_{i\in \mathcal{I}} \omega(b_i^*\cdot b_i)=1 \end{equation}
Remark: $\mathcal{I}$ is a priori not supposed to be countable but $\sum_{i\in \mathcal{I}} \omega(b_i^*\cdot b_i)=1 $ is a bounded sum of positive numbers, and one can show that only countable many summands are non-zero.
Question: i want to show that the folium of a state is closed in the norm of $\mathfrak{A}^*$, the Banach space dual of $\mathfrak{A}$, i.e. $ \left\lVert \omega \right\rVert_{\mathfrak{A}}:= sup_{\left\lVert a\right\rVert=1} |\omega(a)|$
(This statement holds for the usual definition of folium, but in fact I originally wanted to check that the folium of a state as defined here is the folium of the GNS representation of that state.)
Ideas? a state $b\cdot \omega$ with $b$ such that $\omega(b^*b)=1$ is in the folium and this latter is generated by convex linear combinations of such states, i.e. is it the convex hull of such states. I checked that one cannot conclude from this that the folium is closed, but maybe one can say something if it is a subset of the set of all states $S(\mathfrak{A})$ which is known to be weak-* compact, in particular norm closed.