Hilbert $C^*$-modules over $\varSigma ^*$-algebras

Clifford A. Bearden¹

¹ Department of Mathematics University of Houston Houston, TX 77204-3008, U.S.A.

Studia Mathematica, Tome 235 (2016) no. 3, pp. 269-304

Résumé

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.

Détail

DOI : 10.4064/sm8616-9-2016

Keywords: emph varsigma * algebra concrete * algebra sequentially closed weak operator topology study appropriate class * modules varsigma * algebras analogous class * modules selfdual * modules * algebras able obtain varsigma * versions virtually results basic theory * * modules second half paper study modules possessing weak sequential form condition being countably generated particular highlight paper varsigma * module completion varsigma * analogue selfdual completion * module * algebra which has elegant uniqueness condition countably generated