Fr\'echet algebras with a dominating Hilbert algebra norm

TL;DR

This paper characterizes certain commutative Fréchet *-subalgebras of a noncommutative algebra of unbounded operators, showing many classical function algebras are contained within it, revealing their topological structure.

Contribution

It provides a simple characterization of unital commutative Fréchet *-subalgebras of the algebra of unbounded operators, linking them to nuclear power series spaces and showing their inclusion of classical function algebras.

Findings

Many natural Fréchet *-algebras are subalgebras of re9chet algebra

Characterization of subalgebras as nuclear power series spaces

Inclusion of smooth function algebras like C^\u2200(M) and Schwartz space alculus

Abstract

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 Fr\'echet ${}^*$-subalgebras of $\mathscr{L}^*(s)$ isomorphic as a Fr\'echet spaces to nuclear power series spaces $\Lambda_\infty(\alpha)$ of infinite type. It appears that many natural Fr\'echet ${}^*$-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 $\mathbb{R}^n$.