Automorphisms of $\mathscr{P}(λ)/\mathscr{I}_κ$

We study conditions on automorphisms of Boolean algebras of the form $P(λ)/I_κ$ (where $λ$ is an uncountable cardinal and $I_κ$ is the ideal of sets of cardinality less than $κ$) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every cardinality-preserving automorphism of $P(2^κ)/I_{κ^+}$ which is trivial on all sets of cardinality $κ^+$ is trivial, and that $MA_{\aleph_1}$ implies that every automorphism of $P(\mathbb{R})/Fin$ is trivial on a cocountable set.

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