State of $l^{\infty}(G)$

Question

Recently I've come across the definition of an amenable group, which says the following:

"A countable discrete group $G$ is amenable if there exist a state $\mu$ on $l ^{\infty}(G)$ which is invariant under the left translation action."

What does "state" mean in this context? Does it mean some operator from $l ^{\infty}(G)$ to $l ^{\infty}(G)$?

Thank you

Answer 1

A state is a norm 1 positive linear functional. So it is a linear map $\phi:l^\infty(G)\rightarrow\mathbb{C}$ with $\phi(\textbf{1}) = 1$ and $\phi(f) > 0$ whenever $f(g) > 0$ $\forall g\in G$.