Localized strict topologies on multiplier algebras of pro-$C^*$-algebras
Keywords:
pro- $C^*$-algebra, $\sigma$-$C^*$-algebra, locally convex, seminorm, polar, absolutely convex, barrel, bornological, bounded set, localization, strict topology, multiplier algebra, Tychonoff space, completely regular, completely Hausdorff, functionally Hausdorff, compactly generated, $\kappa$-space, adjunction, limit, colimit, compactification, ultrafilterAbstract
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- $C^*$-algebra 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.
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