Topologies related to (I)-envelopes

We investigate the question whether the (I)-envelope of any subset of a dual to a Banach space $X$ may be described as the closed convex hull in a suitable topology. If $X$ contains no copy of ${\mathrm{\ell }}^1$ then the weak topology generated by functionals of the first Baire class in the weak${}^{\ast }$ topology works. On the other hand, if $X$ contains a complemented copy of ${\mathrm{\ell }}^1$ or $X=C([0,1])$ no locally convex topology works. If we do not require the topology to be locally convex, the problem is still open. We further introduce and compare several natural intermediate closure operators on a dual Banach space. Finally, we collect several intringuing open problems related to (I)-envelopes.