  We introduce the notion of the $\alpha$-Haagerup approximation property for
$\alpha\in[0,1/2]$ using a one-parameter family of positive cones studied by
Araki and show that the $\alpha$-Haagerup approximation property actually does
not depend on a choice of $\alpha$. This gives us a direct proof of the fact
that two characterizations of the Haagerup approximation property are
equivalent, one in terms of the standard form and the other in terms of
completely positive maps.
  We also discuss the $L^p$-Haagerup approximation property for a
non-commutative $L^p$-spaces associated with a von Neumann algebra
($1<p<\infty$) and show the independency of the $L^p$-Haagerup approximation
property on $p$.
