57 research outputs found
On the behaviour of Brauer -dimensions under finitely-generated field extensions
The present paper shows that if or , where is the set of prime numbers, then there exist characteristic fields , of Brauer dimension Brd and
infinite absolute Brauer -dimensions abrd, for all not dividing . This ensures that Brd, , for every finitely-generated transcendental
extension . We also prove that each sequence , , satisfying the conditions and , equals the sequence abrd, , for a field of characteristic zero.Comment: LaTeX, 14 pages: published in Journal of Algebra {\bf 428} (2015),
190-204; the abstract in the Metadata updated to fit the one of the pape
Integrable and Chaotic Systems Associated with Fractal Groups
Fractal groups (also called self-similar groups) is the class of groups
discovered by the first author in the 80-s of the last century with the purpose
to solve some famous problems in mathematics, including the question raising to
von Neumann about non-elementary amenability (in the association with studies
around the Banach-Tarski Paradox) and John Milnor's question on the existence
of groups of intermediate growth between polynomial and exponential. Fractal
groups arise in various fields of mathematics, including the theory of random
walks, holomorphic dynamics, automata theory, operator algebras, etc. They have
relations to the theory of chaos, quasi-crystals, fractals, and random
Schr\"odinger operators. One of important developments is the relation of them
to the multi-dimensional dynamics, theory of joint spectrum of pencil of
operators, and spectral theory of Laplace operator on graphs. The paper gives a
quick access to these topics, provide calculation and analysis of
multi-dimensional rational maps arising via the Schur complement in some
important examples, including the first group of intermediate growth and its
overgroup, contains discussion of the dichotomy "integrable-chaotic" in the
considered model, and suggests a possible probabilistic approach to the study
of discussed problems.Comment: 48 pages, 15 figure
Operator Algebras and Abstract Classification
This dissertation is dedicated to the study of operator spaces, operator algebras, and their automorphisms using methods from logic, particularly descriptive set theory and model theory. The material is divided into three main themes. The first one concerns the notion of Polish groupoids and functorial complexity. Such a study is motivated by the fact that the categories of Elliott-classifiable algebras, Elliott invariants, abelian separable C*-algebras, and arbitrary separable C*-algebras have the same complexity according to the usual notion of Borel complexity. The goal is to provide a functorial refinement of Borel complexity, able to capture the complexity of classifying the objects in a functorial way. Our main result is that functorial Borel complexity provides a finer distinction of the complexity of functorial classification problems.
The second main theme concerns the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory. Our results show that, for any non-elementary simple separable C*-algebra, the problem of classifying its automorphisms up to unitary equivalence transcends countable structures. Furthermore we prove that in the unital case the relation of unitary equivalence obeys the following dichotomy: it is either smooth, when the algebra has continuous trace, or not classifiable by countable structures.
The last theme concerns applications of model theory to the study and construction of interesting operator spaces and operator systems. Specifically we show that the Gurarij operator space introduced by Oikhberg can be characterized as the Fraisse limit of the class of finite-dimensional 1-exact operator spaces. This proves that the Gurarij operator space is unique, homogeneous, and universal among separable 1-exact operator spaces. Moreover we prove that, while being 1-exact, the Gurarij operator space does not embed into any exact C*-algebra. Furthermore the ternary ring of operators generated by the Gurarij operator space is canonical, and does not depend on the concrete representation chosen. We also construct the operator system analog of the Gurarij operator space, and prove that it has analogous properties
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