Automatic continuity and amenability in the non-commutative Schwartz space

Abstract

We show that all positive functionals as well as all derivations of the noncommutative Schwartz space are continuous. We also prove this space is not boundedly approximately amenable however it is approximately amenable.

One-sided ideals of the non-commutative Schwartz space

“…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 object has received reasonable attention recently, mostly due to Ciaś-see [3], Domański-see [7] and the author-see [21,22]. This paper is a continuation of the previous work.…”

“…it admits a Hölder functional calculus-see [3,Th. 5.2] and all positive functionals as well as all derivations are automatically continuous-see [21,Ths. 11,13].…”

“…On the other hand it causes some technical difficulties, e.g. it has neither a bounded approximate identity-see [21,Prop. 2] nor is it a locally C * -algebra-see [13,Ths.…”

“…They are devoted only to one-sided ideals since S is topologically simple, i.e. there are no non-trivial closed two-sided ideals-see [21,Th. 4].…”

Commutative subalgebras of the algebra of smooth operators

“…The algebra L(s , s) is isomorphic as a Fréchet * -algebra to the algebra In these forms, the algebra L(s , s) usually appears and plays a significant role in K -theory of Fréchet algebras (see Bhatt [16][17][18][19] and his forthcoming paper "The noncommutative Schwartz space is weakly amenable"). Moreover, in the context of algebras of unbounded operators, the algebra L(s , s) appears in the book [20] as …”