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Noncommutative topological dynamics
Author(s) -
Rai Timothy
Publication year - 2016
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdw011
Subject(s) - noncommutative geometry , mathematics , group (periodic table) , pure mathematics , transitive relation , semigroup , order (exchange) , topological dynamics , topological group , ideal (ethics) , orthogonality , noncommutative algebraic geometry , automorphism , topology (electrical circuits) , noncommutative quantum field theory , combinatorics , topological tensor product , biochemistry , chemistry , philosophy , geometry , organic chemistry , finance , epistemology , functional analysis , economics , gene
We study group actions on C* ‐algebras by interpreting dynamical phenomena K ‐theoretically, that is, by looking at the induced actions on the K 0 ‐group and the Cuntz semigroup of the algebra. A K ‐theoretic filling condition is shown to characterize minimal C* ‐dynamical systems, whereas topological freeness is described by an algebraic orthogonality property. We introduce a notion of topological transitivity in the noncommutative setting that extends the classical definition and characterizes prime reduced crossed products. Such conditions are used to study the order ideal structure of the K 0 groups as well as generate examples of minimal, topologically free, and topologically transitive actions of discrete groups on noncommutative AF‐algebras.