自相似群、代数与Schur补
Self-similar groups, algebras and Schur complements
摘要：
本文的第一部分介绍了自相似群上的$C^*$ -代数，研究了它们的性质以及与已知代数的关系。这些代数被构造为与群的自相似性相关的Cuntz-Pimsner代数（及其同态像）的子代数。研究了与代数相关的核性、单性和Morita等价性等性质。第二部分研究自相似代数元素的Schur补变换。我们研究了这种变换的性质，并将其应用于自相似$C^*-$代数中Markov型元的谱问题。这与群和图上的离散拉普拉斯算子的谱问题有关。舒尔补方法在许多情况下的应用减少了研究多维有理变换的不变集（通常是"奇怪吸引子“的类型）的谱问题。提供了许多说明性的例子。最后，我们观察了自相似群上随机游动的Schur补变换与Bartholdi-Kaimanovich-Virag变换之间的关系.
英文摘要：
In the first part of the article we introduce $C^*$-algebras associated to self-similar groups and study their properties and relations to known algebras. The algebras are constructed as sub-algebras of the Cuntz-Pimsner algebra (and its homomorphic images) associated with the self-similarity of the group. We study such properties as nuclearity, simplicity and Morita equivalence with algebras related to solenoids. The second part deals with the Schur complement transformations of elements of self-similar algebras. We study properties of such transformations and apply them to the spectral problem for Markov type elements in self-similar $C^*-$algebras. This is related to the spectral problem of the discrete Laplace operator on groups and graphs. Application of the Schur complement method in many situations reduces the spectral problem to study of invariant sets (very often of the type of a ``strange attractor'') of a multidimensional rational transformation. A number of illustrating examples is provided. Finally we observe a relation between the Schur complement transformations and Bartholdi-Kaimanovich-Virag transformations of random walks on self-similar groups.
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