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

Zemer Kosloff

doi:10.1007/s11856-013-0069-9

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

Zemer Kosloff^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

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 III_λ 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.

Original language	American English
Pages (from-to)	485-506
Number of pages	22
Journal	Israel Journal of Mathematics
Volume	199
Issue number	1
DOIs	https://doi.org/10.1007/s11856-013-0069-9
State	Published - 1 Jan 2014
Externally published	Yes

Bibliographical note

Funding Information:
∗This research was was supported by The Israel Science Foundation grant No. 1114/08. Received May 9, 2012 and in revised form October 31, 2012

Publisher Copyright:
© 2014, Hebrew University Magnes Press.

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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{\textquoteright}s and Weiss{\textquoteright}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.",

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AB - 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.

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

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