$\varphi$-contractibility and $\varphi$-Connes amenability coincide with some older notions

Abstract

It is shown that various definitions of $\varphi$-Connes amenability as introduced independently in \cite{Gh-Ja, mah, Sh-Am}, are just rediscovering existing notions and presenting them in different ways. It is also proved that even $\varphi$-contractibility as defined in \cite{Sangani}, is equivalent to an older and simpler concept.