Isometries of certain operator algebras

To a given basis $\phi _1,\dotsc ,\phi _n$ on an $n$-dimensional Hilbert space $\mathcal H$, we associate the algebra $\mathfrak A$ of all linear operators on $\mathcal H$ having every $\phi _j$ as an eigenvector. So, $\mathfrak A$ is commutative, semisimple, and $n$-dimensional. Given two algebras of this type, $\mathfrak A$ and $\mathfrak B$, there is a natural algebraic isomorphism $\tau$ of $\mathfrak A$ and $\mathfrak B$. We study the question: When does $\tau$ preserve the operator norm?