Is it consistent with ZFC that there exists an automorphism of $P(\omega\_1)/\mathrm{NS}\_{\omega\_1}$ which is not the identity?



Consistency of the nonrigidity of $P(\omega_1)/NS$
It is consistent relative to large cardinals that $P(\omega\_1)/NS \cong \mathcal B(\mathrm{Col}(\omega,\omega\_1))$ and many other homogeneous algebras.


It almost looks like the answer is yes in $L$, but I'm not sure. In $L$ there are no precipitous ideals, and GCH holds. A [theorem of Balcar and Franek](http://ams.rice.edu/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=balcar%20AND%20franek&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=884452) implies that therefore $\mathcal P(\omega\_1)/NS$ is forcing-equivalent to $\mathrm{Col}(\omega,\omega\_2)$. Moreover, by general facts about collapsing, there is a dense embedding of $\mathrm{Col}(\omega,\omega\_2)$ into $\mathcal P(\omega\_1)/NS$. $\mathrm{Col}(\omega,\omega\_2)$ carries many nontrivial isomorphisms, but as Yair pointed out, we may not necessarily be able to lift this to an algebra strictly smaller than the completion.


