Isomorphisms of standard operator algebras

Let X and Y be Banach spaces, $\dim X = \infty$, and let $\mathcal{A}$ and $\mathcal{B}$ be standard operator algebras on X and Y, respectively. Assume that $\phi :\mathcal{A} \to \mathcal{B}$ is a bijective mapping satisfying $\left\Vert {\phi (AB) - \phi (A)\phi (B)} \right\Vert \leq \varepsilon ,A,B \in \mathcal{A}$, where $\varepsilon$ is a given positive real number (no linearity or continuity of $\phi$ is assumed). Then $\phi$ is a spatially implemented linear or conjugate linear algebra isomorphism. In particular, $\phi$ is continuous.