Computable presentations of reduced group C*-algebras of groups with the rapid decay property

We show that whenever $G$ is a finitely generated group with the rapid decay property (property RD), then the norm on the reduced group C*-algebra $C_r^{\ast }(G)$ is computable if (and only if) $G$ has solvable word problem. We also give an example of a finitely presented group $G$ with solvable word problem for which the norm on $C_r^{\ast }(G)$ is not computable.