Self-similar groups, operator algebras and Schur complement
Rostislav Grigorchuk - Texas A&M University, College Station, Texas, United States (email) Abstract:
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 subalgebras 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.
Keywords: self-similar groups,
self-similarity, Cuntz-Pimsner algebra, Schur complement,
renormalization, Markov operator, spectrum, matrix recursion,
hyperbolic dynamics, rational maps, Moore diagram, Mealy automaton,
amenability, random walks, Munchhausen trick, self-affinemeasure,
Grigorchuk group, iteratedmonodromy groups, limit space.
Received: March 2007; Revised: April 2007; Available Online: April 2007. |