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.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let $L^1(K)$ be the usual Lebesgue space of $K$ with respect to the left Haar measure. We investigate some properties of $L^1(K)$ under a locally convex topology $\beta ^1$. Among other things, the semireflexivity of $(L^1(K), \beta ^1)$ and of sequentially$\beta ^1$-continuous functionals is studied. We also show that $(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if $K$ is compact.