Symmetry and inverse closedness for Banach $^*$-algebras associated to discrete groups

Abstract

A discrete group $\mathrm{G}$ is called rigidly symmetric if for every $C^*$-algebra $\mathcal{A}$ the projective tensor product $\ell^1(\mathrm{G})\widehat\otimes A$ is a symmetric Banach $^*$-algebra. For such a group we show that the twisted crossed product $\ell^1_{\alpha,\omega}(\mathrm{G};\mathcal{A})$ is also a symmetric Banach $^*$-algebra, for every twisted action $(\alpha,\omega)$ of $\mathrm{G}$ in a $C^*$-algebra $\mathcal{A}$. We extend this property to other types of decay, replacing the $\ell^1$-condition. We also make the connection with certain classes of twisted kernels, used in a theory of integral operators involving group $2$-cocycles. The algebra of these kernels is studied, both in intrinsic and in represented version.