A+ CATEGORY SCIENTIFIC UNIT

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

Volume 235 / 2016

Clifford A. Bearden Studia Mathematica 235 (2016), 269-304 MSC: 46L08, 46L05, 46H25. DOI: 10.4064/sm8616-9-2016 Published online: 21 October 2016

Abstract

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.

Authors

  • Clifford A. BeardenDepartment of Mathematics
    University of Houston
    Houston, TX 77204-3008, U.S.A.
    e-mail

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