The Jacobson radical for analytic crossed products

Abstract

We characterise the (Jacobson) radical of the analytic crossed product of C_0(X) by the non-negative integers (Z_+), answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the radical of analytic crossed products of C_0(X) by (Z_+)^d. The radical consists of all elements whose `Fourier coefficients' vanish on the recurrent points of the dynamical system (and the first one is zero). The multi-dimensional version requires a variation of the notion of recurrence, taking into account the various degrees of freedom.