Hilbert C*-modules from group actions: beyond the finite orbits case

Michael Frank, Vladimir Manuilov, and Evgenij Troitsky. "Hilbert C*-modules from group actions: beyond the finite orbits case." Studia Mathematica 200.2 (2010): 131-148. <http://eudml.org/doc/285739>.

@article{MichaelFrank2010,
abstract = {Continuous actions of topological groups on compact Hausdorff spaces X are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows one to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For Lyapunov stable actions the derived invariant mean $M(ϕ_\{x\})$ is continuous on X for any ϕ ∈ C(X), and the induced C*-valued inner product corresponds to a conditional expectation from C(X) onto the fixed-point algebra of the action defined by averaging on orbits. In the case of self-duality of the Hilbert C*-module all orbits are shown to have the same cardinality. Stable actions on compact metric spaces give rise to C*-reflexive Hilbert C*-modules. The same is true if the cardinality of finite orbits is uniformly bounded and the number of closures of infinite orbits is finite. A number of examples illustrate typical situations appearing beyond the classified cases.},
author = {Michael Frank, Vladimir Manuilov, Evgenij Troitsky},
journal = {Studia Mathematica},
keywords = {continuous action; topological group; Hilbert -module; -reflexivity; self-duality; -algebra},
language = {eng},
number = {2},
pages = {131-148},
title = {Hilbert C*-modules from group actions: beyond the finite orbits case},
url = {http://eudml.org/doc/285739},
volume = {200},
year = {2010},
}

TY - JOUR
AU - Michael Frank
AU - Vladimir Manuilov
AU - Evgenij Troitsky
TI - Hilbert C*-modules from group actions: beyond the finite orbits case
JO - Studia Mathematica
PY - 2010
VL - 200
IS - 2
SP - 131
EP - 148
AB - Continuous actions of topological groups on compact Hausdorff spaces X are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows one to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For Lyapunov stable actions the derived invariant mean $M(ϕ_{x})$ is continuous on X for any ϕ ∈ C(X), and the induced C*-valued inner product corresponds to a conditional expectation from C(X) onto the fixed-point algebra of the action defined by averaging on orbits. In the case of self-duality of the Hilbert C*-module all orbits are shown to have the same cardinality. Stable actions on compact metric spaces give rise to C*-reflexive Hilbert C*-modules. The same is true if the cardinality of finite orbits is uniformly bounded and the number of closures of infinite orbits is finite. A number of examples illustrate typical situations appearing beyond the classified cases.
LA - eng
KW - continuous action; topological group; Hilbert -module; -reflexivity; self-duality; -algebra
UR - http://eudml.org/doc/285739
ER -