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 $C(X{)}_1^{\prime }$ 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 $C(X{)}_{\ast }^{\prime }$ of $C(X)$ in $C^{\ast }(\Sigma )$, the enveloping $C^{\ast }$-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 $C(X{)}_{\ast }^{\prime }$ 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 $C^{\ast }$-algebra is $C(X{)}_{\ast }^{\prime }$. Furthermore, we establish necessary and sufficient conditions for projections onto $C(X{)}_1^{\prime }$ and $C(X{)}_{\ast }^{\prime }$ 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.

arXiv commentSome typos corrected. Final version, 23 pages, to appear in Studia Mathematica

References in corpus (1)

• On the Banach $\ast$-algebra crossed product associated with a topological dynamical system

Cited by in corpus (4)

• Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system
• Amenable crossed product Banach algebras associated with a class of ${\mathrm{C}}^{\ast }$-dynamical systems
• Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system
• Topologically irreducible representations of the Banach *-algebra associated with a dynamical system