  We introduce a new equivalence relation on groups, which we call von Neumann
equivalence, that is coarser than both measure equivalence and
$W^*$-equivalence. We introduce a general procedure for inducing actions in
this setting and use this to show that many analytic properties, such as
amenability, property (T), and the Haagerup property, are preserved under von
Neumann equivalence. We also show that proper proximality, which was defined
recently by Boutonnet, Ioana, and the second author using dynamics, is also
preserved under von Neumann equivalence. In particular, proper proximality is
preserved under both measure equivalence and $W^*$-equivalence, and from this
we obtain examples of non-inner amenable groups that are not properly proximal.
