A Gelfand-Naimark type theorem

TL;DR

This paper generalizes the Gelfand-Naimark theorem by constructing the spectrum of a specific subalgebra of continuous bounded functions on a space, linking algebraic properties to topological features of the spectrum.

Contribution

It introduces a simple construction of the spectrum as an open subspace of the Stone-Cech compactification, enabling analysis of its topological properties based on the algebra and space.

Findings

Characterizes when the spectrum is connected or locally connected.

Identifies conditions for the spectrum to be pseudocompact or extremally disconnected.

Establishes criteria for the spectrum to be an F-space or basically disconnected.

Abstract

Let $X$ be a completely regular space. For a non-vanishing self-adjoint Banach subalgebra $H$ of $C_B(X)$ which has local units we construct the spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of the Stone-Cech compactification of $X$ which contains $X$ as a dense subspace. The construction of $\mathfrak{sp}(H)$ is simple. This enables us to study certain properties of $\mathfrak{sp}(H)$, among them are various compactness and connectedness properties. In particular, we find necessary and sufficient conditions in terms of either $H$ or $X$ under which $\mathfrak{sp}(H)$ is connected, locally connected and pseudocompact, strongly zero-dimensional, basically disconnected, extremally disconnected, or an $F$-space.