Characterization of commutative algebras embedded into the algebra of smooth operators

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 $s$.