When is the strict topology bornological?

Question

Let $X$ be a completely regular Hausdorff space. Are there known conditions under which the algebra of bounded continuous functions on $X$, endowed with the strict topology, is bornological?

(Of course, I am not interested in easy conditions such as $X$ being compact - in which the strict topology reduces to the uniform topology.)

I already know of a somewhat related result (obtained independently by Shirota and Nachbin in 1954): the algebra of continuous (not necessarily bounded!) functions on $X$ endowed with the compact-open topology is bornological if and only if $X$ is realcompact. Does a similar result exist for the strict topology and bounded functions?

Answer 1

There are several strict topologies on the space $C_b(T)$ of bounded continuous functions on $T$: $\beta_t$, $\beta_\tau$, ... The following is valid.

The topology $\beta_t$ or $\beta_\tau$ (resp., $\beta_s$, $\beta_p$, or $\beta_\sigma$) is bornological iff $T$ is compact (resp. pseudocompact).

The proof can be found in: R. F. Wheeler, A survey of Baire measures and strict topologies, Expositiones Math. 2(1983), 97-190.