Let $ \mathbb{F}_2$ denotes the free group generated by a,b, denote this group by $G$. Then consider the von Neumann algebra $L(G)$ generated by the family $\{L_{x_g} : g \in G\}$, here, with $g \in G$,we denote by $x_g$ the function on $G$ that takes the value 1 at g and 0 at other elements of $G$. Then, note that we have the following relations:

$(L_{x_g})^*=L_{(x_g)^{-1}} , L_{x_g}L_{x_h}=L_{x_g*x_h}=L_{x_{gh}}$, then, for any $ A \in L(G),$we can set $ A=\sum_{g \in g}\mu_g L_{x_g},$ with $\mu_g \in \mathbb{C}.$

When we calculate $ ||Ax_{h}||^2 $, we find that $ \sum_{g \in G}|\mu_g|^2 < \infty.$

Then, is this condition sufficient for $ A \in L(G) $? Or some stronger condition is necessary? Like $ \sum_{g \in G}|\mu_g| < \infty,$ or something else?

Let $\mathbb{F}_2$ denotes the free group generated by a,b, denote this group by $G$. Then consider the von Neumann algebra $L(G)$ generated by the family $\{L_{x_g} : g \in G\}$, here, with $g \in G$, we denote by $x_g$ the function on $G$ that takes the value 1 at g and 0 at other elements of $G$. Then, note that we have the following relations:

$(L_{x_g})^*=L_{(x_g)^{-1}} , L_{x_g}L_{x_h}=L_{x_g*x_h}=L_{x_{gh}}$, then, for any $ A \in L(G),$we can set $ A=\sum_{g \in g}\mu_g L_{x_g},$ with $\mu_g \in \mathbb{C}.$

When we calculate $ ||Ax_{h}||^2 $, we find that $ \sum_{g \in G}|\mu_g|^2 < \infty.$

Then, is this condition sufficient for $ A \in L(G) $? Or some stronger condition is necessary? Like $ \sum_{g \in G}|\mu_g| < \infty,$ or something else?