The significance of the space \(s\) of rapidly decreasing sequences in the theory of nuclear Fréchet spaces cannot be overestimated.
In fact, by the Kōmura-Kōomura theorem, a Fréchet space is nuclear if and only if it is isomorphic to some closed subspace of \(s^{\mathbb{N}}\).
Moreover, \(s\) is isomorphic as a Fréchet space to many classical spaces of smooth functions, for example it is isomorphic to the Schwartz space
\(\mathcal S(\mathbb R)\) of smooth rapidly decreasing functions on the real line.
In this talk we deal with some noncommutative analogue of the algebra \(s\), namely with the Fréchet \({}^*\)-algebra \(\mathcal{L}(s',s)\)
of so-called smooth operators. We focus on classification and characterization of closed commutative \({}^*\)-subalgebras of \(\mathcal{L}(s',s)\).
In particular, we show a surprising fact that a closed commutative \({}^*\)-subalgebra of \(\mathcal{L}(s',s)\) is isomorphic as a
Fréchet \({}^*\)-algebra to some closed \({}^*\)-subalgebra of \(s\) if and only if it is isomorphic as a Fréchet space to some complemented subspace
of \(s\). Moreover, we provide an example of a closed commutative \({}^*\)-subalgebra of \(\mathcal{L}(s',s)\) which is not embedded isomorphically into \(s\)
as a closed \({}^*\)-subalgebra.
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