Characterization of commutative algebras embedded into the algebra of smooth operators

Abstract

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

14 pages