An asymptotic definition of K-groups of automorphisms and a non-Bernoullian counter-example |
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Authors: | Robert M Burton Jr |
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Institution: | (1) Department of Mathematics, University of Colorado, 80309 Boulder, Colorado, USA |
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Abstract: | Summary Let G Zn be a group of measure preserving transformations of a Lebesgue space. J. P. Conze 1] has developed an entropy theory for such groups and described a class of groups obeying a form of the Kolmogorov zero-one law called K-groups. A Bernoulli group is a group isomorphic to the group of translates (shifts) of elements of the space
with product measure where X
g
=X is a probability space. Bernoulli groups are also K-groups. Katznelson and Weiss 3] have shown entropy is a complete invariant for isomorphism classes of Bernoulli groups. We give an asymptotic definition of K-groups in terms of finite -algebras and justify this definition in terms of entropy and Conze's formulation. This definition s used to help us construct a K-group G Z
n
that is completely non-Bernoulli, that is one that contains no Bernoulli subgroup. |
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