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

TL;DR

This paper proves that Maharam extensions of certain Bernoulli shifts are K transformations, providing a negative answer to longstanding questions about the existence of specific types of Bernoulli shifts in ergodic theory.

Contribution

It establishes that Maharam extensions of conservative, non-singular K Bernoulli shifts without an a.c.i.p. are K transformations, clarifying their type classification.

Findings

Maharam extension of a conservative, non-singular K Bernoulli shift is a K transformation.

Such shifts are either of type II_1 or type III_1.

Negative answer to questions about existence of type II_∞ or type III_λ Bernoulli shifts.

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.