An Algebra of Continuous Functions as a Continuous Envelope of Its Subalgebras

S. S. Akbarov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 2, pp. 75-77

Résumé

To an arbitrary involutive stereotype algebra $A$ the continuous envelope operation assigns its nearest, in some sense, involutive stereotype algebra $\operatorname{\sf{Env}}_{\mathcal C}A$ so that homomorphisms to various $C^*$-algebras separate the elements of $\operatorname{\sf{Env}}_{\mathcal C}A$ but do not distinguish between the properties of $A$ and those of $\operatorname{\sf{Env}}_{\mathcal C}A$. If $A$ is an involutive stereotype subalgebra in the algebra ${\mathcal C}(M)$ of continuous functions on a paracompact locally compact topological space $M$, then, for ${\mathcal C}(M)$ to be a continuous envelope of $A$, i.e., $\operatorname{\sf{Env}}_{\mathcal C}A={\mathcal C}(M)$, it is necessary but not sufficient that $A$ be dense in ${\mathcal C}(M)$. In this note we announce a necessary and sufficient condition for this: the involutive spectrum of $A$ must coincide with $M$ up to a weakening of the topology such that the system of compact subsets in $M$ and the topology on each compact subset remains the same.

Détail

Keywords: $C^*$-algebra, stereotype algebra.