Hilbert 𝐶*-modules in which all closed submodules are complemented

Magajna, Bojan

doi:10.1090/S0002-9939-97-03551-X

Hilbert $C^*$-modules in which all closed submodules are complemented
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Proc. Amer. Math. Soc. 125 (1997), 849-852 Request permission

Abstract:

Let $B$ be a C$^*-$algebra. If there exists a full Hilbert $B-$module $X$ such that $X=Y\oplus Y_\perp$ for each closed submodule $Y$, then $B$ is *-isomorphic to a C$^*-$algebra of (not neccesarily all) compact operators on a Hilbert space.

References

William Arveson, An invitation to $C^*$-algebras, Graduate Texts in Mathematics, No. 39, Springer-Verlag, New York-Heidelberg, 1976. MR 0512360, DOI 10.1007/978-1-4612-6371-5
Bruce A. Barnes, G. J. Murphy, M. R. F. Smyth, and T. T. West, Riesz and Fredholm theory in Banach algebras, Research Notes in Mathematics, vol. 67, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 668516
Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 463928, DOI 10.2140/pjm.1977.71.349
Kjeld Knudsen Jensen and Klaus Thomsen, Elements of $KK$-theory, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991. MR 1124848, DOI 10.1007/978-1-4612-0449-7
E. C. Lance, Hilbert C$^*-$modules, London Math. Soc Lecture Note Series 210, Cambridge Univ. Press, Cambridge, 1995.
Hua Xin Lin, Injective Hilbert $C^\ast$-modules, Pacific J. Math. 154 (1992), no. 1, 131–164. MR 1154736, DOI 10.2140/pjm.1992.154.131
J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1971), 263–269. MR 276734, DOI 10.1007/BF02771592
Gerard J. Murphy, $C^*$-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990. MR 1074574
William L. Paschke, Inner product modules over $B^{\ast }$-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. MR 355613, DOI 10.1090/S0002-9947-1973-0355613-0
Marc A. Rieffel, Induced representations of $C^{\ast }$-algebras, Advances in Math. 13 (1974), 176–257. MR 353003, DOI 10.1016/0001-8708(74)90068-1