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

Abstract

A discrete group $\G$ is called rigidly symmetric if for every $C^{\ast }$-algebra $\A$ the projective tensor product $\ell^1(\G)\widehat\otimes\A$ is a symmetric Banach ${}^{\ast }$-algebra. For such a group we show that the twisted crossed product $\ell^1_{α,ø}(\G;\A)$ is also a symmetric Banach ${}^{\ast }$-algebra, for every twisted action $(\alpha ,ø)$ of $\G$ in a $C^{\ast }$-algebra $\A$\,. 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.

19 pages

Cited by in corpus (1)

• Inverse-closedness of the set of integral operators with $L_1$-continuously varying kernels