Seminormed $\ast$-subalgebras of $\ell^{\infty}(X)$

TL;DR

This paper explores the structure of seminormed *-subalgebras within b(X), focusing on the Gelfand spectrum, the role of the supremum norm, and the measure-theoretic properties of b(X) subalgebras, especially in relation to topology and measure lifting.

Contribution

It characterizes seminormed *-subalgebras of b(X), emphasizing the supremum norm's significance and analyzing measure lifting and topological relations in the Gelfand spectrum.

Findings

The supremum norm uniquely characterizes b(X) as a Banach algebra.

Lifting positive measures from b(X) to the Gelfand spectrum reveals an unexpected support shift.

The topology of X influences the Gelfand spectrum's topology in the case of Borel algebras.

Abstract

Arbitrary representations of a commutative unital ($\ast$-) $\mathbb{F}$-algebra $A$ as a subalgeba of $\mathbb{F}^X$ are considered, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$ and $X\neq\emptyset$. The Gelfand spectrum of $A$ is explained as a topological extension of $X$ where a seminorm on the image of $A$ in $\mathbb{F}^X$ is present. It is shown that among all seminormes, the $\sup$-norm is of special importance which reduces $\mathbb{F}^X$ to $\ell^{\infty}(X)$. The Banach subalgebra of $\ell^{\infty}(X)$ of all $\Sigma$-measurable bounded functions on $X$, is studied for which $\Sigma$ is a $\sigma$-algebra of subsets of $X$. In particular, we study lifting of positive measures from $(X, \Sigma)$ to the Gelfand spectrum of this algebra and observe an unexpected shift in the support of measures. In the case that $\Sigma$ is the Borel algebra of a topology, we study the relation of the underlying topology of $X$ and the one of the Gelfand spectrum.