Commutative subalgebras of the algebra of smooth operators

Abstract

We consider the Fr\'echet ${}^*$-algebra $L(s',s)$ of the so-called smooth operators, i.e. continuous linear operators from the dual $s'$ 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 ${}^*$-subalgebras of $L(s',s)$ which are at the same time isomorphic to closed ${}^*$-subalgebras of $s$ and we provide an example of a closed commutative ${}^*$-subalgebra of $L(s',s)$ which cannot be embedded into $s$.