Symmetric compactifications of the integers and separable quotients of spaces $C_p(X)$

A compactification of the discrete space $ω$ of all integers is called symmetric if it is the quotient space obtained by gluing together the remainders of two copies of some other compactification of $ω$. This is a generalization of both a convergent sequence, which is in a way the minimal symmetric compactification of $ω$, and the Arkhangel'skiĭ--Bereznitskiĭ--Schachermayer space studied in $C_p$-theory, which is in a sense the maximal symmetric compactification of $ω$. We investigate symmetric compactifications of $ω$ and their relations to the Separable Quotient Problem for spaces $C_p(X)$ and to the existence of Josefson--Nissenzweig sequences of finitely supported Borel measures on spaces $X$, in particular with supports of bounded size. Further, we reduce the Separable Quotient Problem for spaces $C_p(K)$, $K$ compact, to the case when $K$ is a totally asymmetric compactification of $ω$. Our results shed some new light on the Grothendieck property of Banach spaces $C(K)$.