Let G be a finitely generated, discrete group. Using a random walk approach, Erschler and Karlsson constructed a homomorphism G ? R. Central to their construction were the word length ? and a well behaved measure ? on G. We consider a modified version of this construction using instead of ? a quasimorphism f of G. Moreover, if a group H acts on G via group automorphisms we show that this technique can be adapted to construct a homomorphism of the semidirect product G ?_? H to R. As an application, we construct a ?-automorphism ?_f of the underlying reduced group C^?-algebra C^?_r(G).

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Random walks on groups, quasimorphisms, *-automorphism