Ergodic characterizations of character amenability and contractibility of Banach algebras (Q657675)

Let \({\mathcal A}\) be a Banach algebra and let \(\phi\in \Delta\)(\({\mathcal A}\)), the spectrum of \({\mathcal A}\) consisting of all non-zero characters on \({\mathcal A}\). \textit{E. Kaniuth, A. T. Lau} and \textit{J. Pym} introduced and studied the concept of \(\phi\)-amenability for Banach algebras in [Math.\ Proc.\ Camb.\ Philos.\ Soc.\ 144, No. 1, 85--96 (2008; Zbl 1145.46027)]. The Banach algebra \({\mathcal A}\) is called \(\phi\)-amenable if there exists a bounded linear functional \(m\) on \({\mathcal A}^{*}\) satisfying \(m(\phi)=1\) and \( m(f\cdot a)=m(f)\phi(a)\) for all \(a\in{\mathcal A}\) and \(f\in{\mathcal A}^*\), where \(f\cdot a\in{\mathcal A}^*\) is defined by \((f\cdot a)(b)=f(ab)\) for all \(b\in{\mathcal A}\). This is a generalization of left amenability of a Lau algebra (or \(F\)-algebra) which was first studied by \textit{A. T.-M. Lau} [Fundam.\ Math.\ 118, 161--175 (1983; Zbl 0545.46051)]. Moreover, the notion of character amenability for Banach algebras was introduced and studied by \textit{M. S. Monfared} in [Math.\ Proc.\ Camb.\ Philos.\ Soc.\ 144, No. 3, 697--706 (2008; Zbl 1153.46029)]; he called \({\mathcal A}\) character amenable if it is \(\phi\)-amenable for all \({\phi}\in \Delta\)(\({\mathcal A}\)) and if it has a bounded right approximate identity. Similarly, the concept of character contractibility has been introduced in [\textit{Z. Hu, M. S. Monfared} and \textit{T. Traynor}, Stud.\ Math.\ 193, No. 1, 53--78 (2009; Zbl 1175.22005)]. Also, alternative characterizations of \(\phi\)-contractibility and \(\phi\)-amenability of \({\mathcal A}\) in terms of projectivity and injectivity of some Banach left \({\mathcal A}\)-modules have been provided in [\textit{R. Nasr-Isfahani} and \textit{S. S. Renani}, Stud.\ Math.\ 202, No. 3, 205--225 (2011; Zbl 1236.46045)]. In the paper under review, the authors introduce and study \(\phi\)-ergodic antirepresentations of \(A\). These antirepresentations are used in order to provide some equivalent conditions for the Banach algebra \({\mathcal A}\) to be \(\phi\)-amenable (\(\phi\)-contractible). Indeed, they extend the results of \textit{R. Nasr-Isfahani} [Bull.\ Iran.\ Math.\ Soc.\ 28, No. 2, 29--35,81 (2002; Zbl 1037.43002)] to the setting of general Banach algebras. Finally, the authors illustrate their obtained results by considering the convolution algebra \(L^p(G)\), where \(1\leq p<\infty\) and \(G\) is a compact group.