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

Abstract

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 $\sp{*}$-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 $C(X)'_1$ 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 $C(X)'_*$ 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 $C(X)'_1$ 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 $X\times\mathbb{T}$. 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 for the periodic points of a homeomorphism of a locally compact Hausdorff space are given.