  Suppose that $T^*$ is an $\omega_1$-Aronszajn tree with no stationary
antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which
preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of
the strong consequences of PFA, such as the failure of very weak club guessing,
that all of the cardinal characteristics of the continuum are greater than
$\omega_1$, and the $P$-ideal dichotomy. On the other hand, PFA($T^*$) implies
some of the consequences of diamond principles, such as the existence of
Knaster forcings which are not stationarily Knaster.
