Dilation invariant Banach limits (Q2196736)

Denote by \(\mathfrak{B}\subseteq \ell_{\infty}^*\) the set of all Banach limits and by \(\mathfrak{B}(C)\) the set of those Banach limits which are invariant under the Cesàro operator \(C\). Let \[ \sigma_n(x_1,x_2,\dots):=(\underbrace{x_1,\dots,x_1}_{n\text{-times}},\underbrace{x_2,\dots,x_2}_{n\text{-times}},\dots), \] i.e., \(\sigma_n\) is the \(n\)-th dilation operator. \(\mathfrak{B}(\sigma_n)\) denotes the set of all Banach limits which are invariant under \(\sigma_n\).\par In [\textit{E. Semenov} et al., Pac. J. Math. 306, No. 1, 357--373 (2020; Zbl 1455.46023)], it was proved that \[ \mathfrak{B}(C)\subseteq \bigcap_{n=2}^{\infty}\mathfrak{B}(\sigma_n)=:\Sigma \] and in [\textit{E. Alekhno} et al., Stud. Math. 242, No.~1, 79--107 (2018; Zbl 1407.46017)], it was shown that \(\mathfrak{B}(C)\neq \mathfrak{B}(\sigma_n)\) for every \(n\geq 2\). In this paper, the authors show that \(\mathfrak{B}(C)\) is a proper subset of \(\Sigma\). They also prove that \(\|B_1-B_2\|=2\) for every \(B_1\in \mathfrak{B}(\sigma_n)\) and every \(B_2\) in the norm-closed convex hull of the extreme points of \(\mathfrak{B}\).