C*-dynamical rapid decay

Some well known results by Haagerup, Jolissaint and de la Harpe may be extended to the setting of a reduced crossed product of a C*-algebra A by a discrete group $G.$ We show that for many discrete groups, which include Gromov's hyperbolic groups and finitely generated discrete groups of polynomial growth, an inequality of the form $$\|X\| \leq C \sqrt{\sum_{g \in G} (1+|g|)^4 \|X_g\|^2 }$$ holds for any finitely supported operator $X$ in the reduced crossed product.