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Abstract

Let $({\sf G},\alpha , \omega ,\mathfrak B)$ be a measurable twisted action of a locally compact group ${\sf G}$ on a Banach $^*$-algebra $\mathfrak B$, and $\mathfrak A$ a differential Banach $^*$-subalgebra of $\mathfrak B$ which is stable under the said action. We observe that $L^1_{\alpha ,\omega }({\sf G},\mathfrak A)$ is a differential subalgebra of $L^1_{\alpha ,\omega }({\sf G},\mathfrak B)$. We use this fact to provide new examples of groups with symmetric Banach $^*$-algebras. In particular, we prove that discrete rigidly symmetric extensions of compact groups are symmetric, and semidirect products ${\sf K}\rtimes {\sf H}$, with ${\sf H}$ symmetric and ${\sf K}$ compact, are symmetric.