Hilbert $C^*$-modules over $\varSigma ^*$-algebras
A $\varSigma ^*$-algebra is a concrete $C^*$-algebra that is sequentially closed in the weak operator topology. We study an appropriate class of $C^*$-modules over $\varSigma ^*$-algebras analogous to the class of $W^*$-modules (selfdual $C^*$-modules over $W^*$-algebras), and we are able to obtain $\varSigma ^*$-versions of virtually all the results in the basic theory of $C^*$- and $W^*$-modules. In the second half of the paper, we study modules possessing a weak sequential form of the condition of being countably generated. A particular highlight of the paper is the “$\varSigma ^*$-module completion,” a $\varSigma ^*$-analogue of the selfdual completion of a $C^*$-module over a $W^*$-algebra, which has an elegant uniqueness condition in the countably generated case.