Monic representations of the Cuntz algebra and Markov measures
Monic representations of the Cuntz algebra and Markov measures
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Cuntz 代数和马尔可夫测度的莫尼克表示
DOI：
10.1016/j.jfa.2014.05.016
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
P. Jorgensen
中科院分区：
文献类型：
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作者：
D. Dutkay;P. Jorgensen
我们研究了Cuntz代数O N的表示。然而，对于固定N，已知O N表示的等价类集合不具有Borel横截面，存在各种可分类的表示子类。我们研究了O N的一元表示，它具有正则阿贝尔子代数的循环向量。我们证明了O - N具有一个包含所有正单胞表示的一定的全称表示。一元表示的一个大的类的例子是基于马尔可夫测度。我们对它们进行了分类，得到了不同参数产生相互奇异的马尔可夫测度，扩展了Kakutani的经典结果。基于Kakutani测度的单调表示正是那些具有一维循环S i -不变空间的表示。
We study representations of the Cuntz algebras O N. While, for fixed N, the set of equivalence classes of representations of O N is known not to have a Borel cross section, there are various subclasses of representations which can be classified. We study monic representations of O N, that have a cyclic vector for the canonical abelian subalgebra. We show that O N has a certain universal representation which contains all positive monic representations. A large class of examples of monic representations is based on Markov measures. We classify them and as a consequence we obtain that different parameters yield mutually singular Markov measure, extending the classical result of Kakutani. The monic representations based on the Kakutani measures are exactly the ones that have a one-dimensional cyclic S i⁎-invariant space.