Commutative subalgebras of the algebra of smooth operators

Abstract

We consider the Fréchet * -algebra L(s ′ , s) ⊆ L(ℓ2) 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.