Commutative subalgebras of the algebra of smooth operators

TL;DR

This paper studies the structure of commutative subalgebras within the algebra of smooth operators, identifying which can be embedded into the algebra of rapidly decreasing sequences and providing examples of those that cannot.

Contribution

It characterizes all closed commutative *-subalgebras of the algebra of smooth operators that are isomorphic to subalgebras of s, and presents an example of a non-embeddable subalgebra.

Findings

Characterization of embeddable commutative *-subalgebras

Existence of non-embeddable commutative *-subalgebras

Connection between smooth operators and sequence algebra s

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 $s$.