  The paper deal with the noncommutative Fr\'echet ${}^*$-algebra
$\mathcal{L}(s',s)$ of the so-called smooth operators, i.e. linear and
continuous operators acting from the space $s'$ of slowly increasing sequences
to the Fr\'echet space $s$ of rapidly decreasing sequences. By a canonical
identification, this algebra of smooth operators can be also seen as the
algebra of the rapidly decreasing matrices. We give a full description of
closed commutative ${}^*$-subalgebras of this algebra and we show that every
closed subspace of $s$ with basis is isomorphic (as a Fr\'echet space) to some
closed commutative ${}^*$-subalgebra of $\mathcal{L}(s',s)$. As a consequence,
we give some equivalent formulation of the long-standing Quasi-equivalence
Conjecture for closed subspaces of $s$.
