Commutative subalgebras of the algebra of smooth operators

Abstract

We consider the Fréchet ${}^{\ast }$-algebra $L(s^{\prime },s)$ of the so-called smooth operators, i.e. continuous linear operators from the dual $s^{\prime }$ of the space $s$ of rapidly decreasing sequences into $s$. This algebra is a non-commutative analogue of the algebra $s$. We characterize all closed commutative ${}^{\ast }$-subalgebras of $L(s^{\prime },s)$ which are at the same time isomorphic to closed ${}^{\ast }$-subalgebras of $s$ and we provide an example of a closed commutative ${}^{\ast }$-subalgebra of $L(s^{\prime },s)$ which cannot be embedded into $s$.