An elementary proof of the characterization of isomorphisms of standard operator algebras

This study provides an elementary proof of the well-known fact that any isomorphism $\pi : \mathcal{A}\to \mathcal{B}$ of standard operator algebras on normed spaces $X , Y$, respectively, is spatial; i.e., there exists a topological isomorphism $T : X \to Y$ such that $\pi(A) = TAT^{-1}$ for any $A \in \mathcal{A}$. In particular, $\pi$ is continuous.