Fréchet algebras with a dominating Hilbert algebra norm

英文摘要：

Let $\mathscr{L}^*(s)$ be the maximal $\mathcal{O}^*$-algebra of unbounded operators on $\ell_2$ whose domain is the space $s$ of rapidly decreasing sequences. This is a noncommutative topological algebra with involution which can be identified, for instance, with the algebra $\mathscr L(s)\cap\mathscr L(s')$ or the algebra of multipliers for the algebra $\mathscr{L}(s',s)$ of smooth compact operators. We give a simple characterization of unital commutative Fréchet ${}^*$-subalgebras of $\mathscr{L}^*(s)$ isomorphic as a Fréchet spaces to nuclear power series spaces $Λ_\infty(α)$ of infinite type. It appears that many natural Fréchet ${}^*$-algebras are closed ${}^*$-subalgebras of $\mathscr{L}^*(s)$, for example, the algebras $C^\infty(M)$ of smooth functions on smooth compact manifolds and the algebra $\mathscr S (\mathbb{R}^n)$ of smooth rapidly decreasing functions on $\mathbb{R}^n$.