The Jacobson Radical for Analytic Crossed Products

We characterise the Jacobson radical of an analytic crossed product C0(X)×φ Z+, answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the Jacobson radical of analytic crossed products C0(X)×φ Zd+. This consists of all elements whose “Fourier coefficients” vanish on the recurrent points of the dynamical system (and the first one is zero). The multidimensional version requires a variation of the notion of recurrence, taking into account the various degrees of freedom.