  We introduce the notion of proper proximality for finite von Neumann
algebras, which naturally extends the notion of proper proximality for groups.
Apart from the group von Neumann algebras of properly proximal groups, we
provide a number of additional examples, including examples in the settings of
free products, crossed products, and compact quantum groups. Using this notion,
we answer a question of Popa by showing that the group von Neumann algebra of a
nonamenable inner amenable group cannot embed into a free group factor. We also
introduce a notion of proper proximality for probability measure preserving
actions, which gives an invariant for the orbit equivalence relation. This
gives a new approach for establishing strong ergodicity type properties, and we
use this in the setting of Gaussian actions to expand on solid ergodicity
results first established by Chifan and Ioana, and later generalized by
Boutonnet. The techniques developed also allow us to answer a problem left open
by Anantharaman-Delaroche in 1995, by showing the equivalence between the
Haagerup property and the compact approximation property for II$_1$ factors.
