Hilbert $C^*$-modules over $Σ^*$-algebras

A $Σ^*$-algebra is a concrete $C^*$-algebra that is sequentially closed in the weak operator topology. We study an appropriate class of $C^*$-modules over $Σ^*$-algebras analogous to the class of $W^*$-modules (selfdual $C^*$-modules over $W^*$-algebras), and we are able to obtain $Σ^*$-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 "$Σ^*$-module completion," a $Σ^*$-analogue of the selfdual completion of a $C^*$-module over a $W^*$-algebra, which has an elegant uniqueness condition in the countably generated case.

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