Seminormed ⁎-subalgebras of ℓ∞(X)

Journal article, 2017

Arbitrary representations of an involutive commutative unital F-algebra A as a subalgebra of FX are considered, where F=C or R and X≠∅. The Gelfand spectrum of A is explained as a topological extension of X where a seminorm on the image of A in FX is present. It is shown that among all seminorms, the sup-norm is of special importance which reduces FX to ℓ∞(X). The Banach subalgebra of ℓ∞(X) of all Σ-measurable bounded functions on X, Mb(X,Σ), is studied for which Σ is a σ-algebra of subsets of X. In particular, we study lifting of positive measures from (X,Σ) to the Gelfand spectrum of Mb(X,Σ) and observe an unexpected shift in the support of measures. In the case that Σ is the Borel algebra of a topology, we study the relation of the underlying topology of X and the topology of the Gelfand spectrum of Mb(X,Σ).