Covariance algebra of a partial dynamical system

Bartosz Kosma Kwaśniewski

Open Mathematics (2005)

  • Volume: 3, Issue: 4, page 718-765
  • ISSN: 2391-5455

Abstract

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A pair (X, α) is a partial dynamical system if X is a compact topological space and α: Δ→ X is a continuous mapping such that Δ is open. Additionally we assume here that Δ is closed and α(Δ) is open. Such systems arise naturally while dealing with commutative C *-dynamical systems. In this paper we construct and investigate a universal C *-algebra C *(X,α) which agrees with the partial crossed product [10] in the case α is injective, and with the crossed product by a monomorphism [22] in the case α is onto. The main method here is to use the description of maximal ideal space of a coefficient algebra, cf. [16, 18], in order to construct a larger system, ( X ˜ , α ˜ ) where α ˜ is a partial homeomorphism. Hence one may apply in extenso the partial crossed product theory [10, 13], In particular, one generalizes the notions of topological freeness and invariance of a set, which are being use afterwards to obtain the Isomorphism Theorem and the complete description of ideals of C *(X, α).

How to cite

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Bartosz Kosma Kwaśniewski. "Covariance algebra of a partial dynamical system." Open Mathematics 3.4 (2005): 718-765. <http://eudml.org/doc/268682>.

@article{BartoszKosmaKwaśniewski2005,
abstract = {A pair (X, α) is a partial dynamical system if X is a compact topological space and α: Δ→ X is a continuous mapping such that Δ is open. Additionally we assume here that Δ is closed and α(Δ) is open. Such systems arise naturally while dealing with commutative C *-dynamical systems. In this paper we construct and investigate a universal C *-algebra C *(X,α) which agrees with the partial crossed product [10] in the case α is injective, and with the crossed product by a monomorphism [22] in the case α is onto. The main method here is to use the description of maximal ideal space of a coefficient algebra, cf. [16, 18], in order to construct a larger system, \[(\tilde\{X\},\tilde\{\alpha \})\] where \[\tilde\{\alpha \}\] is a partial homeomorphism. Hence one may apply in extenso the partial crossed product theory [10, 13], In particular, one generalizes the notions of topological freeness and invariance of a set, which are being use afterwards to obtain the Isomorphism Theorem and the complete description of ideals of C *(X, α).},
author = {Bartosz Kosma Kwaśniewski},
journal = {Open Mathematics},
keywords = {47L65; 46L45; 37B99},
language = {eng},
number = {4},
pages = {718-765},
title = {Covariance algebra of a partial dynamical system},
url = {http://eudml.org/doc/268682},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Bartosz Kosma Kwaśniewski
TI - Covariance algebra of a partial dynamical system
JO - Open Mathematics
PY - 2005
VL - 3
IS - 4
SP - 718
EP - 765
AB - A pair (X, α) is a partial dynamical system if X is a compact topological space and α: Δ→ X is a continuous mapping such that Δ is open. Additionally we assume here that Δ is closed and α(Δ) is open. Such systems arise naturally while dealing with commutative C *-dynamical systems. In this paper we construct and investigate a universal C *-algebra C *(X,α) which agrees with the partial crossed product [10] in the case α is injective, and with the crossed product by a monomorphism [22] in the case α is onto. The main method here is to use the description of maximal ideal space of a coefficient algebra, cf. [16, 18], in order to construct a larger system, \[(\tilde{X},\tilde{\alpha })\] where \[\tilde{\alpha }\] is a partial homeomorphism. Hence one may apply in extenso the partial crossed product theory [10, 13], In particular, one generalizes the notions of topological freeness and invariance of a set, which are being use afterwards to obtain the Isomorphism Theorem and the complete description of ideals of C *(X, α).
LA - eng
KW - 47L65; 46L45; 37B99
UR - http://eudml.org/doc/268682
ER -

References

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