Characterization of commutative algebras embedded into the algebra of smooth operators

The paper deal with the noncommutative Fr'echet ${}^{\ast }$-algebra $\mathcal{L}(s^{\mathrm{\prime }},s)$ of the so-called smooth operators, i.e. linear and continuous operators acting from the space $s^{\mathrm{\prime }}$ 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 ${}^{\ast }$-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 ${}^{\ast }$-subalgebra of $\mathcal{L}(s^{\mathrm{\prime }},s)$. As a consequence, we give some equivalent formulation of the long-standing Quasi-equivalence Conjecture for closed subspaces of $s$.