Symmetry and Inverse Closedness for Some Banach ${}^{\ast }$-Algebras Associated to Discrete Groups

A discrete group $\text{G}$ is called rigidly symmetric if for every $C^{\ast }$-algebra $\text{A}$ the projective tensor product ${\mathrm{\ell }}^1(\text{G})\widehat{\otimes }\text{A}$ is a symmetric Banach ${}^{\ast }$-algebra. For such a group we show that the twisted crossed product ${\mathrm{\ell }}_{\alpha ,\text{o}}^1(\text{G};\text{A})$ is also a symmetric Banach ${}^{\ast }$-algebra, for every twisted action $(\alpha ,\text{o})$ of $\text{G}$ in a $C^{\ast }$-algebra $\text{A}$,. We extend this property to other types of decay, replacing the ${\mathrm{\ell }}^1$-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.