  The notion of von Neumann equivalence (vNE), which encapsulates both measure
equivalence and $W^*$-equivalence, was introduced recently by Jesse Peterson,
Lauren Ruth and the author. They showed that many analytic properties, such as
amenability, property (T), the Haagerup property, and proper proximality are
preserved under von Neumann equivalence. In this article, we expand on the list
of properties that are stable under von Neumann equivalence, and prove that
weak amenability, weak Haagerup property, and the approximation property (AP)
are von Neumann equivalence invariants. In particular, we get that (AP) is
stable under measure equivalence. Furthermore, our techniques give an alternate
proof for vNE-invariance of the Haagerup property.
