We show in a straightforward way that the non-commutative Schwartz space is weakly amenable. At the end, we leave an open problem.

It is an open question whether every derivation of a Fréchet GB$^{\ast }$-algebra $A[{\it\tau}]$ is continuous. We give an affirmative answer for the case where $A[{\it\tau}]$ is a smooth Fréchet nuclear GB$^{\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB$^{\ast }$-algebras which are not pro-C$^{\ast }$-algebras.

We show that every continuous self-adjoint functional on the noncommutative Schwartz space can be decomposed into a difference of two positive functionals. Moreover, this decomposition is minimal in the natural sense.

Towards an involutive analogue of a result on the semisimplicity of ${\ell }^{1} (S)$ by Hewitt and Zuckerman, we show that, given an abelian $\ast $-semigroup $S$, the commutative convolution Banach $\ast $-algebra ${\ell }^{1} (S)$ is $\ast $-semisimple if and only if Hermitian bounded semicharacters on $S$ separate the points of $S$; and we search for an intrinsic separation property on $S$ equivalent to $\ast $-semisimplicity. Very many natural involutive analogues of Hewitt and Zuckerman’s separation property are shown not to work, thereby exhibiting intricacies involved in analysis on $S$.

A hermitian algebra is a unital associative ℂ-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in ℝ. In the case of an algebra 𝒜 endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open and the inversion mapping is continuous, we construct the smooth structures on the appropriate versions of flag manifolds. Then we prove that if such a locally convex algebra 𝒜 is endowed with a continuous involution, then it is a hermitian algebra if and only if the natural action of all unitary groups Un(𝒜) on each flag manifold is transitive.