Maximal abelian subalgebras and projections in two Banach algebras associated with a topological dynamical system

Marcel de Jeu, and Jun Tomiyama. "Maximal abelian subalgebras and projections in two Banach algebras associated with a topological dynamical system." Studia Mathematica 208.1 (2012): 47-75. <http://eudml.org/doc/285682>.

@article{MarceldeJeu2012,
abstract = {If Σ = (X,σ) is a topological dynamical system, where X is a compact Hausdorff space and σ is a homeomorphism of X, then a crossed product Banach *-algebra ℓ¹(Σ) is naturally associated with these data. If X consists of one point, then ℓ¹(Σ) is the group algebra of the integers. The commutant C(X)₁' of C(X) in ℓ¹(Σ) is known to be a maximal abelian subalgebra which has non-zero intersection with each non-zero closed ideal, and the same holds for the commutant C(X)'⁎ of C(X) in C*(Σ), the enveloping C*-algebra of ℓ¹(Σ). This intersection property has proven to be a valuable tool in investigating these algebras. Motivated by this pivotal role, we study C(X)₁' and C(X)'⁎ in detail in the present paper. The maximal ideal space of C(X)₁' is described explicitly, and is seen to coincide with its pure state space and to be a topological quotient of X×𝕋. We show that C(X)₁' is hermitian and semisimple, and that its enveloping C*-algebra is C(X)'⁎. Furthermore, we establish necessary and sufficient conditions for projections onto C(X)₁' and C(X)'⁎ to exist, and give explicit formulas for such projections, which we show to be unique. In the appendix, topological results on the periodic points of a homeomorphism of a locally compact Hausdorff space are given.},
author = {Marcel de Jeu, Jun Tomiyama},
journal = {Studia Mathematica},
keywords = {involutive Banach algebra; crossed product; maximal abelian subalgebra; pure state space; norm one projection; positive projection; topological dynamical system},
language = {eng},
number = {1},
pages = {47-75},
title = {Maximal abelian subalgebras and projections in two Banach algebras associated with a topological dynamical system},
url = {http://eudml.org/doc/285682},
volume = {208},
year = {2012},
}

TY - JOUR
AU - Marcel de Jeu
AU - Jun Tomiyama
TI - Maximal abelian subalgebras and projections in two Banach algebras associated with a topological dynamical system
JO - Studia Mathematica
PY - 2012
VL - 208
IS - 1
SP - 47
EP - 75
AB - If Σ = (X,σ) is a topological dynamical system, where X is a compact Hausdorff space and σ is a homeomorphism of X, then a crossed product Banach *-algebra ℓ¹(Σ) is naturally associated with these data. If X consists of one point, then ℓ¹(Σ) is the group algebra of the integers. The commutant C(X)₁' of C(X) in ℓ¹(Σ) is known to be a maximal abelian subalgebra which has non-zero intersection with each non-zero closed ideal, and the same holds for the commutant C(X)'⁎ of C(X) in C*(Σ), the enveloping C*-algebra of ℓ¹(Σ). This intersection property has proven to be a valuable tool in investigating these algebras. Motivated by this pivotal role, we study C(X)₁' and C(X)'⁎ in detail in the present paper. The maximal ideal space of C(X)₁' is described explicitly, and is seen to coincide with its pure state space and to be a topological quotient of X×𝕋. We show that C(X)₁' is hermitian and semisimple, and that its enveloping C*-algebra is C(X)'⁎. Furthermore, we establish necessary and sufficient conditions for projections onto C(X)₁' and C(X)'⁎ to exist, and give explicit formulas for such projections, which we show to be unique. In the appendix, topological results on the periodic points of a homeomorphism of a locally compact Hausdorff space are given.
LA - eng
KW - involutive Banach algebra; crossed product; maximal abelian subalgebra; pure state space; norm one projection; positive projection; topological dynamical system
UR - http://eudml.org/doc/285682
ER -