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We consider the Fréchet \({}^*\)-algebra \({\mathcal {L}}(s',s)\subseteq \mathcal L(\ell _2)\) of the so-called smooth operators, i.e. continuous linear operators from the dual \(s'\) of the space s of rapidly decreasing sequences to s. This algebra is a non-commutative analogue of the algebra s. We characterize closed \({}^*\)-subalgebras of \({\mathcal {L}}(s',s)\) which are at the same time isomorphic to closed \({}^*\)-subalgebras of s and we provide an example of a closed commutative \({}^*\)-subalgebra of \({\mathcal {L}}(s',s)\) which cannot be embedded into s.

Commutative subalgebras of the algebra of smooth operators