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Abstract

We investigate $2$-local Jordan automorphisms on operator algebras. In particular, we show that every $2$-local Jordan automorphism of the algebra of all $n\times n$ real or complex matrices is either an automorphism or an anti-automorphism. The same is true for $2$-local Jordan automorphisms of any subalgebra of $\mathcal B$ which contains the ideal of all compact operators on $X$, where $X$ is a real or complex separable Banach spaces and $\mathcal B$ is the algebra of all bounded linear operators on $X$.