Complete embeddings of the Cohen algebra into three families of c.c.c., non-measurable Boolean algebras.
The question is: which ccc forcings necessarily add Cohen reals? It is known that measure algebras do not. This paper shows that three classes of ccc forcings which are not measure algebras do add Cohen reals: Galvin-Hajnal algebras (in fact, they add cf \(2^{\omega}\) many), Argyros algebras, and the atomless Gaifman algebras (or their atomless parts). In some sense these results are negative: von Neumann conjectured that a Boolean \(\sigma\)-algebra with ccc and the weak \((\omega,\omega)\)-distributive law is a measure algebra. The completions of the Galvin-Hajnal, Argyros, and Gaifman algebras are complete ccc algebras which fail to satisfy certain conditions that measure algebras satisfy, hence were candidates for counterexamples. But the results of this paper show that the weak \((\omega,\omega)\)-distributive law fails in these algebras.
scientific article; zbMATH DE number 2101151