Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Seminormed $\ast$-subalgebras of $\ell^{\infty}(X)$   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.

29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con