Right Coideal Subalgebras, Hopf Galois Objects, and $\imath$-Hopf Algebras

We study right comodule algebras and their connections with Hopf bi-Galois objects and right coideal subalgebras. For a Hopf $2$-cocycle $σ$ on $K$ and a unital right-colinear algebra map $Γ:A\to K$, we show that $Γ:A_σ\to K^σ$ is an algebra map precisely when $σ$ is trivial on $\operatorname{im}Γ\o