Glasnik Matematicki, Vol. 56, No. 2 (2021), 343-374. \( \)

HILBERT \(C^{*}\)-MODULES IN WHICH ALL RELATIVELY STRICTLY CLOSED SUBMODULES ARE COMPLEMENTED

Boris Guljaš

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
e-mail:guljas@math.hr

Abstract.   We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

2020 Mathematics Subject Classification.   46L08, 46L05.

Key words and phrases.   Hilbert modules, orthogonal complementing, reletive strict topology for modules, hereditary subalgebras and hereditary modules, direct sums of algebras and modules

https://doi.org/10.3336/gm.56.2.08

References:

1. Lj. Arambašić, Another characterization of Hilbert C*-modules over compact operators, J. Math. Anal. Appl. 344 (2008), 735–740.
   MathSciNet    CrossRef

2. Lj. Arambašić, D. Bakić, Frames and outer frames for Hilbert C*-modules, Linear Multilinear Algebra 65 (2017), 381–431.
   MathSciNet    CrossRef

3. W. Arveson, An invitation to C*-algebras, Springer, New York, 1976.
   MathSciNet

4. D. Bakić, B. Guljaš, Hilbert C*-modules over C*-algebras of compact operators, Acta Sci. Math. (Szeged) 68 (2002), 249–269.
   MathSciNet

5. D. Bakić, B. Guljaš, On a class of module maps of Hilbert C*-modules, Math. Commun. 7 (2002), 177–192.
   MathSciNet

6. D. Bakić, B. Guljaš, Extensions of Hilbert C*-modules, Houston J. Math. 30 (2004), 537–558.
   MathSciNet

7. D. Bakić, B. Guljaš, Extensions of Hilbert modules II, Glas. Mat. Ser. III 38 (2003), 341–357.
   MathSciNet    CrossRef

8. B. Gramsch, Eine Idealstruktur Banachscher Operatoralgebren, J. Reine Angew. Math. 225 (1967), 97–115.
   MathSciNet    CrossRef

9. B. Guljaš, Orthogonal complementing in Hilbert modules, Ann. Funct. Anal. 10 (2019), 196–202.
   MathSciNet    CrossRef

10. E. Luft, The two-sided closed ideals of the algebra of bounded linear operators of a Hilbert space, Czechoslovak Math. J. 18 (1968), 595–605.
    MathSciNet

11. B. Magajna, Hilbert modules in which all closed submodules are complemented, Proc. Amer. Math. Soc. 125 (1997), 849–852.
    MathSciNet    CrossRef

12. G. J. Murphy, C*-algebras and operator theory, Academic Press, Inc., Boston, 1990.
    MathSciNet

13. I. Raeburn and D. P. Williams, Morita equivalence and continuous-trace algebras, AMS, Providence, 1998.
    MathSciNet    CrossRef

14. J. Schweizer, A description of Hilbert C*-modules in which all closed submodules are orthogonally closed, Proc. Amer. Math. Soc. 127 (1999), 2123–2125.
    MathSciNet    CrossRef

15. N. E. Wegge-Olsen, K-theory and algebras, Oxford University Press, Oxford, 1993.
    MathSciNet