C*-crossed products and shift spaces
Silvestrov, Sergei; Carlsen, Toke Meier (2007). C*-crossed products and shift spaces. Expositiones Mathematicae, 25, (4), 275 - 307
|
Published
|
English
Authors:
Silvestrov, Sergei
;
Carlsen, Toke Meier
Department:
Mathematics (Faculty of Engineering)
Non-commutative Geometry-lup-obsolete
Project:
Non-commutative Geometry in Mathematics and Physics
Non-commutative Analysis of Dynamics, Fractals and Wavelets
Research Group:
Non-commutative Geometry-lup-obsolete
Abstract:
We use Exel's C*-crossed products associated to non-invertible dynamical systems to associate a C*-algebra to arbitrary shift space. We show that this C*-algebra is canonically isomorphic to the C*-algebra associated to a shift space given by Carlsen [Cuntz–Pimsner C*-algebras associated with subshifts, Internat. J. Math. (2004) 28, to appear, available at arXiv:math.OA/0505503], has the
C*-algebra defined by Carlsen and Matsumoto [Some remarks on the C*-algebras associated with subshifts, Math. Scand. 95 (1) (2004) 145–160] as a quotient, and possesses properties indicating that it can be thought of as the universal C*-algebra associated to a shift space.
We also consider its representations and its relationship to other C*-algebras associated to shift spaces. We show that it can be viewed as a generalization of the universal Cuntz–Krieger algebra, discuss uniqueness and present a faithful representation, show that it is nuclear and satisfies the Universal Coefficient Theorem, provide conditions for it being simple and purely infinite, show that the constructed C*-algebras and thus their K-theory, K0 and K1, are conjugacy invariants of one-sided shift spaces, present formulas for those invariants, and present a description of the structure of gauge invariant ideals.
Keywords:
Invariants ;
K-theory ;
Dynamical systems ;
Shift spaces ;
Cuntz–Krieger algebras ;
C*-algebra
Cite this