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?