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

Marcel de Jeu ¹   ; Jun Tomiyama ²

¹ Mathematical Institute Leiden University P.O. Box 9512 2300 RA Leiden, The Netherlands
² Department of Mathematics Tokyo Metropolitan University Minami-Osawa, Hachioji City, Japan

Studia Mathematica, Tome 208 (2012) no. 1, pp. 47-75

Résumé

If $\Sigma=(X, \sigma)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed product Banach $^{\ast}$-algebra $\ell^1(\Sigma)$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1(\Sigma)$ is the group algebra of the integers. The commutant $C(X)_1^\prime$ of $C({X})$ in $\ell^1({\Sigma})$ 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 $C(X)^\prime_\ast$ of $C({X})$ in $C^*(\Sigma)$, the enveloping $C^*$-algebra of $\ell^1(\Sigma)$. This intersection property has proven to be a valuable tool in investigating these algebras. Motivated by this pivotal role, we study $C(X)_1^\prime$ and $C(X)^\prime_\ast$ in detail in the present paper. The maximal ideal space of $C(X)_1^\prime$ is described explicitly, and is seen to coincide with its pure state space and to be a topological quotient of $X\times\mathbb T$. We show that $C(X)_1^\prime$ is hermitian and semisimple, and that its enveloping $C^*$-algebra is $C(X)^\prime_\ast$. Furthermore, we establish necessary and sufficient conditions for projections onto $C(X)_1^\prime$ and $C(X)^\prime_\ast$ 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.

Détail

DOI : 10.4064/sm208-1-4

Keywords: sigma sigma topological dynamical system where compact hausdorff space sigma homeomorphism crossed product banach ast algebra ell sigma naturally associated these consists point ell sigma group algebra integers commutant prime ell sigma known maximal abelian subalgebra which has non zero intersection each non zero closed ideal holds commutant prime ast * sigma enveloping * algebra ell sigma intersection property has proven valuable tool investigating these algebras motivated pivotal role study prime prime ast detail present paper maximal ideal space prime described explicitly seen coincide its pure state space topological quotient times mathbb prime hermitian semisimple its enveloping * algebra prime ast furthermore establish necessary sufficient conditions projections prime prime ast exist explicit formulas projections which unique appendix topological results periodic points homeomorphism locally compact hausdorff space given