## Annals of Functional Analysis

### Orthogonal complementing in Hilbert $C^{*}$-modules

Boris Guljaš

#### Abstract

We characterize orthogonally complemented submodules in Hilbert $C^{*}$-modules by their orthogonal closures. Applying Magajna’s characterization of Hilbert $C^{*}$-modules over $C^{*}$-algebras of compact operators by the complementing property of submodules, we give an elementary proof of Schweizer’s characterization of Hilbert $C^{*}$-modules over $C^{*}$-algebras of compact operators. Also, we prove analogous characterization theorems for $C^{*}$-algebras of compact operators related to topological properties of submodules of strict completions of Hilbert modules over a nonunital $C^{*}$-algebra.

#### Article information

Source
Ann. Funct. Anal., Volume 10, Number 2 (2019), 196-202.

Dates
Accepted: 19 October 2018
First available in Project Euclid: 19 March 2019

https://projecteuclid.org/euclid.afa/1552960868

Digital Object Identifier
doi:10.1215/20088752-2018-0028

Mathematical Reviews number (MathSciNet)
MR3941381

Zentralblatt MATH identifier
07083888

#### Citation

Guljaš, Boris. Orthogonal complementing in Hilbert $C^{*}$ -modules. Ann. Funct. Anal. 10 (2019), no. 2, 196--202. doi:10.1215/20088752-2018-0028. https://projecteuclid.org/euclid.afa/1552960868

#### References

• [1] D. Bakić and B. Guljaš, On a class of module maps of Hilbert $C^{*}$-modules, Math. Commun. 7 (2002), no. 2, 177–192.
• [2] D. Bakić and B. Guljaš, Extensions of Hilbert $C^{*}$-modules II, Glas. Mat. Ser. III 38(58) (2003), no. 2, 341–357.
• [3] D. Bakić and B. Guljaš, Extensions of Hilbert $C^{*}$-modules, Houston J. Math. 30 (2004), no. 2, 537–558.
• [4] R. Gebhardt and K. Schmüdgen, Unbounded operators on Hilbert $C^{*}$-modules, Internat. J. Math. 26 (2015), no. 11, art. ID 1550094.
• [5] B. Magajna, Hilbert $C^{*}$-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc. 125 (1997), no. 3, 849–852.
• [6] I. Raeburn and D. P. Williams, Morita equivalence and continuous-trace $C^{*}$-algebras, Math. Surveys Monogr. 60, Amer. Math. Soc., Providence, 1998.
• [7] J. Schweizer, A description of Hilbert $C^{*}$-modules in which all closed submodules are orthogonally closed, Proc. Amer. Math. Soc. 127 (1999), no. 7, 2123–2125.
• [8] N. E. Wegge-Olsen, K-theory and $C^{*}$-algebras: A Friendly Approach, Oxford Univ. Press, New York, 1993.