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

Abstract

We show that the Maharam extension of a type III, conservative and nonsingular K Bernoulli 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∞ or type IIIλ with λ ≠ 1 Bernoulli shift. A conservative non-singular K, in the sense of Silva and Thieullen, Bernoulli shift is either of type II1 or of type III1.