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Selfsimilar groups, operator algebras and Schur complement
1.  Texas A&M University, College Station, Texas, United States, United States 
The second part deals with Schur complement transformations of elements of selfsimilar algebras. We study the properties of such transformations and apply them to the spectral problem for Markov type elements in selfsimilar $C$*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 Schur complement transformations and BartholdiKaimanovichVirag transformations of random walks on selfsimilar groups.
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2019 Impact Factor: 0.465
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