Hilbert \(C^*\)-modules over \(\varSigma ^*\)-algebras (Q2836001)

\textit{E. B. Davies} [Commun. Math. Phys. 8, 147--161 (1968; Zbl 0153.44701)] introduced the notion of a \(\Sigma^*\)-algebra as a concrete \(C^*\)-algebra that is sequentially closed in the weak operator topology (WOT). The class of \(\Sigma^*\)-algebras can be regarded between the classes of \(C^*\)-algebras and \(W^*\)-algebras. Inspired by the paper of \textit{M. Hamana} [Int. J. Math. 3, No. 2, 185--204 (1992; Zbl 0810.46047)], the author of the present paper introduces an appropriate class of \(C^*\)-modules over \(\Sigma^*\)-algebras analogous to the class of \(W^*\)-modules (self-dual \(C^*\)-modules over \(W^*\)-algebras) and obtains \(\Sigma^*\)-versions of general results in the basic theory of \(C^*\)- and \(W^*\)-modules. Among other things, he shows that, in analogy with the situation in \(C^*\)-module theory and \(W^*\)-module theory, \(\Sigma^*\)-modules are essentially the same as WOT sequentially closed TROs and essentially the same as corners of \(\Sigma^*\)-algebras. He proves that the \(\Sigma^*\)-module completion of a \(C^*\)-module over a \(\Sigma^*\)-algebra is in an analogy with the self-dual completion of a \(C^*\)-module over a \(W^*\)-algebra. In addition, he studies the \(\Sigma^*\)-module completion as a \(\Sigma^*\)-analogue of the self-dual completion of a \(C^*\)-module over a \(W^*\)-algebra, which has a nice uniqueness condition in the countably generated case.