To a given basis
ϕ
1
,
…
,
ϕ
n
\phi _1,\dotsc ,\phi _n
on an
n
n
-dimensional Hilbert space
H
\mathcal H
, we associate the algebra
A
\mathfrak A
of all linear operators on
H
\mathcal H
having every
ϕ
j
\phi _j
as an eigenvector. So,
A
\mathfrak A
is commutative, semisimple, and
n
n
-dimensional. Given two algebras of this type,
A
\mathfrak A
and
B
\mathfrak B
, there is a natural algebraic isomorphism
τ
\tau
of
A
\mathfrak A
and
B
\mathfrak B
. We study the question: When does
τ
\tau
preserve the operator norm?