A SIMPLE MAP WITH NO PRIME FACTORS

An ergodic measure-preserving transformation T of a probability space is said to be simple (of order 2) if every ergodic joining lambda of T with itself is either mu x mu or an off-diagonal measure mu(S), i.e., mu(S)(A x B) = mu(A boolean AND S-nB) for some invertible, measure pr eserving S commuting with T. Veech proved that if T is simple then T i s a group extension of any of its non-trivial factors. Here we constru ct an example of a weakly mixing simple T which has no prime factors. This is achieved by constructing an action of the countable Abelian gr oup Z + G, where G = +(infinity)(i=1) Z(2) such that the Z-subaction i s simple and has centralizer coinciding with the Full Z + G-action.