  Let $\mathscr{L}^*(s)$ be the maximal $\mathcal{O}^*$-algebra of unbounded
operators on $\ell_2$ whose domain is the space $s$ of rapidly decreasing
sequences. This is a noncommutative topological algebra with involution which
can be identified, for instance, with the algebra $\mathscr L(s)\cap\mathscr
L(s')$ or the algebra of multipliers for the algebra $\mathscr{L}(s',s)$ of
smooth compact operators. We give a simple characterization of unital
commutative Fr\'echet ${}^*$-subalgebras of $\mathscr{L}^*(s)$ isomorphic as a
Fr\'echet spaces to nuclear power series spaces $\Lambda_\infty(\alpha)$ of
infinite type. It appears that many natural Fr\'echet ${}^*$-algebras are
closed ${}^*$-subalgebras of $\mathscr{L}^*(s)$, for example, the algebras
$C^\infty(M)$ of smooth functions on smooth compact manifolds and the algebra
$\mathscr S (\mathbb{R}^n)$ of smooth rapidly decreasing functions on
$\mathbb{R}^n$.
