[Paper Review] On closed subalgebras of CB(X)

Majid Farhadi, M. R. Koushesh|arXiv (Cornell University)|Dec 31, 2017

Advanced Topics in Algebra7 references3 citations

TL;DR

This paper establishes a new representation of the Gelfand spectrum of a non-vanishing, self-adjoint, closed subalgebra $ H $ of $ C_B(X) $, where $ X $ is a completely regular space, by constructing the spectrum $ \mathfrak{sp}(H) $ as an open subspace of the compactification $ \alpha_HX $ generated by $ H $. The key result shows that $ \mathfrak{sp}(H) $ is separable metrizable if and only if $ H $ is countably generated.

ABSTRACT

For a completely regular space $X$ and a non-vanishing self-adjoint closed subalgebra $H$ of $C_B(X)$ which separates points from closed sets in $X$ we construct the Gelfand spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of the compactification of $X$ generated by $H$. The simple construction of $\mathfrak{sp}(H)$ enables easier examination of its properties. We illustrate this by an example showing that the space $\mathfrak{sp}(H)$ is separable metrizable if and only if $H$ is countably generated.

Motivation & Objective

• To replace the ad hoc assumption of local units in prior work with the standard topological condition of separating points from closed sets.
• To provide a canonical construction of the Gelfand spectrum $ \mathfrak{sp}(H) $ as an open subspace of the $ H $-generated compactification $ \alpha_HX $.
• To enable easier analysis of spectral properties by using a more natural and structured compactification.
• To establish a precise characterization of when the spectrum is separable and metrizable.
• To provide concrete examples of spaces and algebras satisfying the conditions for the theorem to apply.

Proposed method

• Define $ \alpha_HX $ as the closure of the evaluation map $ e: X \to \prod_{h \in H} \overline{h(X)} $, yielding a compactification of $ X $.
• Extend each $ h \in H $ continuously to $ h_\alpha \in C(\alpha_HX) $, ensuring $ h_\alpha|_X = h $.
• Construct the Gelfand spectrum $ \mathfrak{sp}(H) $ as the open subset of $ \alpha_HX $ where all $ h_\alpha $ are non-zero.
• Use the fact that $ \alpha_HX $ is a subspace of a product of compact sets, and when $ H $ is countably generated, $ \alpha_HX $ is separable and metrizable.
• Apply the standard Gelfand–Naimark duality to identify $ \mathfrak{sp}(H) $ as the spectrum of $ H $, with $ H \cong C_0(\mathfrak{sp}(H)) $.
• Leverage density of countable subalgebras to show that if $ H $ is countably generated, then $ \alpha_HX $ is separable metrizable, hence $ \mathfrak{sp}(H) $ inherits these properties.

Experimental results

Research questions

• RQ1 Under what conditions can the Gelfand spectrum of a closed subalgebra $ H \subset C_B(X) $ be naturally embedded in a compactification of $ X $?
• RQ2 How does the topological structure of the spectrum $ \mathfrak{sp}(H) $ relate to the algebraic generation of $ H $?
• RQ3 Can the assumption of local units in earlier representations be replaced by a more standard topological condition?
• RQ4 When is the spectrum $ \mathfrak{sp}(H) $ separable and metrizable?
• RQ5 What classes of spaces and algebras satisfy the point-separating-from-closed-sets condition necessary for the construction?

Key findings

• The Gelfand spectrum $ \mathfrak{sp}(H) $ of a non-vanishing, self-adjoint, closed subalgebra $ H \subset C_B(X) $ that separates points from closed sets is naturally realized as an open subspace of the $ H $-generated compactification $ \alpha_HX $.
• The spectrum $ \mathfrak{sp}(H) $ is separable and metrizable if and only if $ H $ is countably generated.
• The construction of $ \alpha_HX $ as the closure of the evaluation map ensures that all elements of $ H $ extend continuously to $ \alpha_HX $, and $ \mathfrak{sp}(H) $ is the set of points in $ \alpha_HX $ where no $ h_\alpha $ vanishes.
• When $ H $ is countably generated, the algebra $ H $ contains a countable dense subalgebra $ Q $, and $ \alpha_QX $ is a separable metrizable space, hence $ \mathfrak{sp}(H) $ inherits this structure.
• The spectrum $ \mathfrak{sp}(H) $ is homeomorphic to the structure space of $ H $, and $ H \cong C_0(\mathfrak{sp}(H)) $, confirming the duality.
• Examples of such algebras include $ H = \{ f \in C_B(X) : |f|^{-1}([\epsilon,\infty)) \text{ has property } \mathscr{P} \} $, where $ \mathscr{P} $ is a closed-hereditary, countable-union-preserving topological property and $ X $ is locally $ \mathscr{P} $.