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