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Commutative subalgebras of the algebra of smooth operators (1503.07095v1)
Published 24 Mar 2015 in math.FA
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 $s$.