Fréchet algebras with a dominating Hilbert algebra norm

Abstract

Let ${\mathcal{L}}^{\ast }(s)$ be the maximal ${\mathcal{O}}^{\ast }$-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 $\mathcal{L}(s)\cap \mathcal{L}(s^{\prime })$ or the algebra of multipliers for the algebra $\mathcal{L}(s^{\prime },s)$ of smooth compact operators. We give a simple characterization of unital commutative Fréchet ${}^{\ast }$-subalgebras of ${\mathcal{L}}^{\ast }(s)$ isomorphic as a Fréchet spaces to nuclear power series spaces ${\Lambda }_{\infty }(\alpha )$ of infinite type. It appears that many natural Fréchet ${}^{\ast }$-algebras are closed ${}^{\ast }$-subalgebras of ${\mathcal{L}}^{\ast }(s)$, for example, the algebras $C^{\infty }(M)$ of smooth functions on smooth compact manifolds and the algebra $\mathcal{S}({\mathbb{R}}^n)$ of smooth rapidly decreasing functions on ${\mathbb{R}}^n$.

International Conference held at the University of Oulu, July 3-11, 2017

References in corpus (1)

• The multiplier algebra of the noncommutative Schwartz space