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

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