Operational 2-local automorphisms/derivations

Let φ: A→ A be a (not necessarily linear, additive or continuous) map of a standard operator algebra. Suppose for any a,b∈ A there is an algebra automorphism θa,b of A such that {align*} φ(a)φ(b) = θa,b(ab). {align*} We show that either φ or -φ is a linear Jordan homomorphism. Similar results are obtained when any of the following conditions is satisfied: {align*} φ(a) + φ(b) &= θa,b(a+b), \\ φ(a)φ(b)+φ(b)φ(a) &= θa,b(ab+ba), \\ φ(a)φ(b)φ(a) &= θa,b(aba). {align*} We also show that a map φ: M→ M of a semi-finite von Neumann algebra M is a linear derivation if for every a,b∈ M there is a linear derivation Da,b of M such that φ(a)b + aφ(b) = Da,b(ab).