Extreme points of the set of Banach limits

Let ${mathbb{N}}$ be the set of all natural numbers and ${ell_infty=ell_infty (mathbb{N})}$ be the Banach space of all bounded sequences x = (x ₁, x ₂ . . .) with the norm $$|x|_{infty}=sup_{ninmathbb{N}}|x_n|,$$ and let ${ell_infty^*}$ be its Banach dual. Let ${mathfrak{B} subset ell_infty^*}$ be the set of all normalised positive translation invariant functionals (Banach limits) on ? _∞ and let ${ext(mathfrak{B})}$ be the set of all extreme points of ${mathfrak{B}}$ . We prove that an arbitrary sequence (B _j ) _{j ≥ 1}, of distinct points from the set ${ext(mathfrak{B})}$ is 1-equivalent to the unit vector basis of the space ? ₁ of all summable sequences. We also study Cesáro-invariant Banach limits. In particular, we prove that the norm closed convex hull of ${ext(mathfrak{B})}$ does not contain a Cesáro-invariant Banach limit.