On the K property for Maharam extensions of Bernoulli shifts and a question of Krengel

Abstract

We show that the Maharam extension of a conservative. non singular K Bernoulli shift without an a.c.i.p. is a K transformation. This together with the fact that the Maharam extension of a conservative transformation is conservative gives a negative answer to Krengel's and Weiss's questions about existence of a type II}_\infty or type III}_\lambda with \lambda not equal to 1 Bernoulli shift. A conservative non singular K Bernoulli shift is either of type II_1 or of type III_1.