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It is an important result of \v Semrl which states that every 2-local automorphism of the full operator algebra over a separable Hilbert space is necessarily an automorphism. In this paper we strengthen that result quite substantially for *-automorphisms. Indeed, we show that one can compress the defining two equations of 2-local *-automorphisms into one single equation, hence weakening the requirement significantly, but still keeping essentially the conclusion that such maps are necessarily *-automorphisms., Comment: To appear in J. Math. Anal. Appl

We describe the structure of all continuous Jordan triple endomorphisms of the set $\mathbb{P}_2$ of all positive definite $2\times 2$ matrices thus completing a recent result of ours. We also mention an application concerning sorts of surjective generalized isometries on $\mathbb{P}_2$ and, as second application, we complete another former result of ours on the structure of sequential endomorphisms of finite dimensional effect algebras.