Then the Banach $k$-algebras $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi)$, $\widehat{V}_{{\scriptscriptstyle\bullet}}(L|_{Y},\phi|_{Y})$ and $\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})$ are semi-simple. In particular, they are reduced.