One-sided ideals of the non-commutative Schwartz space

Krzysztof Piszczek

doi:10.1007/s00605-015-0758-z

Abstract

We show that maximal one-sided ideals of the non-commutative Schwartz space are closed. We also characterize all closed one-sided ideals of this algebra. As a result, all maximal left ideals are fixed.

Highlights

The problem of characterizing ideals of a given topological algebra belongs to classical ones and there is vast literature devoted to it
The aim of this paper is to address this problem in the case of one particular lmc Fréchet ∗-algebra, the so-called noncommutative Schwartz space
Piszczek to the space C∞(M) of smooth functions on a compact smooth manifold M or to the Schwartz space S(R) of test functions for tempered distributions. This last isomorphism justifies the name for S. This links our object with the structure theory of Fréchet spaces, especially with questions concerning nuclearity or splitting of short exact sequences—see [17, Part IV]

Summary

Introduction

The problem of characterizing ideals of a given topological algebra belongs to classical ones and there is vast literature devoted to it. The aim of this paper is to address this problem in the case of one particular lmc Fréchet ∗-algebra, the so-called noncommutative Schwartz space. This last isomorphism justifies the name for S (the other name algebra of smooth operators—used e.g. by Cias—comes from the first-mentioned isomorphism) This links our object with the structure theory of Fréchet spaces, especially with questions concerning nuclearity or splitting of short exact sequences—see [17, Part IV]. This algebra appears in the context of K-theory—see [4,20] or in the context of cyclic cohomology for crossed products—see [12,23]. The author is very indebted to Tomasz Kania for many valuable comments and discussions and to David Blecher for pointing out the reference [9]

Notation and terminology

Closedness of maximal one-sided ideals

Description of closed one-sided ideals