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Suppose that $T^*$ is an $\omega_1$-Aronszajn tree with no stationaryantichain. We introduce a forcing axiom PFA($T^*$) for proper forcings whichpreserve these properties of $T^*$. We prove that PFA($T^*$) implies many ofthe 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($T^*$) impliessome of the consequences of diamond principles, such as the existence ofKnaster forcings which are not stationarily Knaster.

A forcing axiom for a non-special Aronszajn tree