Is there any example of a complete atomless Boolean algebra with a **non-trivial** abelian automorphism group?


This is equivalent, by Stone duality, to asking for an extremally disconnected compact Hausdorff space with no isolated points, having abelian homeomorphism group. Note that no such example can be metrizable.


I only know of examples of *rigid* complete Boolean algebras, i.e., those having trivial automorphism groups. For examples, see


* Jónsson, B., *A Boolean Algebra Without Proper Automorphisms*, Proc. AMS, 1951.


Another rigid example coming from forcing appears in


* McAloon, K., *Consistency Results About Ordinal Definability*, Ann. Math. Log, 1971.



Complete atomless Boolean algebras with abelian automorphism group
If A is a rigid complete BA, then the automorphism group of AxA is isomorphic to the direct sum of |A| copies of the two-element group.This is proved in the remark following Lemma 1.9 in a paper of McKenzie and Monk (Colloq. Math.
Soc. Janos Bolyai, 1973, 951-988); the result is attributed to de Groot; see the reference above.


