Fixed point properties related to character amenable Banach algebras (Q2802019)

Let \(\varphi\) be a non-zero character on a Banach algebra \(A\). Then \(A\) is called \(\varphi\)-amenable if there exists \(m\in A^{**}\) such that \(m(\varphi)=1\) and \(m(f.a)=\varphi(a)m(f)\) for all \(f\in A^*\) and \(a\in A\), where \(f.a\) is an element of the dual \(A^*\) of \(A\) defined by \((f.a)(b)=f(ab)\) for all \(b\in A\); (cf. \textit{E. Kaniuth} et al. [Math. Proc. Camb. Philos. Soc. 144, No. 1, 85-96 (2008; Zbl 1145.46027)]). In the paper under review, the authors characterize the existence of a \(\varphi\)-mean on certain topological left invariant and left introverted subspaces of \(A^*\) in terms of some common fixed point properties. They describe \(\varphi\)-amenability of \(A\) in terms of the Hahn--Banach extension property. Some applications to the group algebra and the Fourier algebra of a locally compact group are presented as well.