Localized strict topologies on multiplier algebras of pro-$C^*$-algebras

TL;DR

This paper investigates the strict topology on multiplier algebras of pro-$C^*$-algebras, showing it equals a certain localization, and characterizes specific classes of pro-$C^*$-algebras through topological properties and categorical equivalences.

Contribution

It generalizes Taylor's result to bornological pro-$C^*$-algebras and characterizes barreled commutative unital pro-$C^*$-algebras via topological function spaces.

Findings

Strict topology equals its localization on multiplier algebras of bornological pro-$C^*$-algebras.

Characterization of barreled commutative unital pro-$C^*$-algebras as function spaces with specific topologies.

Establishment of a categorical equivalence involving Tychonoff spaces and pro-$C^*$-algebras.

Abstract

The bounded localization $\beta_b$ of a locally convex topology $\beta$ is defined as the finest locally convex topology agreeing with $\beta$ on all bounded sets. We show that the strict topology on the multiplier algebra of a bornological pro-$C^*$-algebras equals its own localization, generalizing the analogous result due to Taylor for multiplier algebras of plain $C^*$-algebras. We also (a) characterize the barreled commutative unital pro-$C^*$-algebras as those of continuous functions on functionally Hausdorff spaces whose relatively pseudocompact subsets are relatively compact, equipped with the topology of uniform convergence on compact subsets, and (b) describe a contravariant equivalence between the category of commutative unital pro-$C^*$-algebras and a category of Tychonoff (rather than functionally Hausdorff) topological spaces.