In this paper we obtain a characterization of finite-dimensional Hilbert C*-modules. It is known that those are the modules for which both underlying C*-algebras are finite-dimensional. We show that such modules can be described by a certain property of bounded sequences of their elements. It turns out that similar property leads to another characterization of Hilbert C*-modules over C*-algebras of compact operators.
References
Lj. Arambaši\' c, Irreducible representations of Hilbert $C^*$-modules, Math. Proc. R. Ir. Acad. 105A (2005), no. 2, 11–24. MR2162903 1098.46045 Lj. Arambaši\' c, Irreducible representations of Hilbert $C^*$-modules, Math. Proc. R. Ir. Acad. 105A (2005), no. 2, 11–24. MR2162903 1098.46045
Lj. Arambaši\' c, Another characterization of Hilbert $C^*$-modules over compact operators, J. Math. Anal. Appl. 344 (2008), 735–740. MR2426303 1149.46046 10.1016/j.jmaa.2008.03.004 Lj. Arambaši\' c, Another characterization of Hilbert $C^*$-modules over compact operators, J. Math. Anal. Appl. 344 (2008), 735–740. MR2426303 1149.46046 10.1016/j.jmaa.2008.03.004
Lj. Arambaši\' c, D. Baki\' c and M.S. Moslehian, A characterization of Hilbert $C^*$-modules over finite dimensional $C^*$-algebras, Oper. Matrices 3 (2009), no. 2, 235–240. MR2522778 1188.46036 Lj. Arambaši\' c, D. Baki\' c and M.S. Moslehian, A characterization of Hilbert $C^*$-modules over finite dimensional $C^*$-algebras, Oper. Matrices 3 (2009), no. 2, 235–240. MR2522778 1188.46036
D. Baki\' c and B. Guljaš, Hilbert $C\sp *$-modules over $C\sp *$-algebras of compact operators, Acta Sci. Math. (Szeged) 68 (2002), no. 1-2, 249–269. MR1916579 D. Baki\' c and B. Guljaš, Hilbert $C\sp *$-modules over $C\sp *$-algebras of compact operators, Acta Sci. Math. (Szeged) 68 (2002), no. 1-2, 249–269. MR1916579
D. Baki\' c and B. Guljaš, On a class of module maps of Hilbert $C^*$-modules, Math. Commun. 7 (2002), no. 2, 177–192. MR1952758 1031.46066 D. Baki\' c and B. Guljaš, On a class of module maps of Hilbert $C^*$-modules, Math. Commun. 7 (2002), no. 2, 177–192. MR1952758 1031.46066
J. Chmieliński, D. Iliševi\' c, M.S. Moslehian and Gh. Sadeghi, Perturbation of the Wigner equation in inner product C*-modules, J. Math. Phys. 49 (2008), no. 3, 033519. MR2406816 1153.81342 J. Chmieliński, D. Iliševi\' c, M.S. Moslehian and Gh. Sadeghi, Perturbation of the Wigner equation in inner product C*-modules, J. Math. Phys. 49 (2008), no. 3, 033519. MR2406816 1153.81342
M. Frank, Characterizing $C^*$-algebras of compact operators by generic categorical properties of Hilbert $C^*$-modules, J. $K$-theory 2 (2008), no. 3, 453–462. MR2465447 M. Frank, Characterizing $C^*$-algebras of compact operators by generic categorical properties of Hilbert $C^*$-modules, J. $K$-theory 2 (2008), no. 3, 453–462. MR2465447
M. Frank and A.A. Pavlov, Banach-Saks properties of C*-algebras and Hilbert $C^*$-modules, Banach J. Math. Anal. 3 (2009), no. 2, 91–102. MR2545176 euclid.bjma/1261086713
M. Frank and A.A. Pavlov, Banach-Saks properties of C*-algebras and Hilbert $C^*$-modules, Banach J. Math. Anal. 3 (2009), no. 2, 91–102. MR2545176 euclid.bjma/1261086713
I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839–858. MR58137 0051.09101 10.2307/2372552 I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839–858. MR58137 0051.09101 10.2307/2372552
B. Magajna, Hilbert $C^*$-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc. 125 (1997), no. 3, 849–852. MR1346981 0865.46038 10.1090/S0002-9939-97-03551-X B. Magajna, Hilbert $C^*$-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc. 125 (1997), no. 3, 849–852. MR1346981 0865.46038 10.1090/S0002-9939-97-03551-X
V.M. Manuilov and E.V. Troitsky, Hilbert $C^*$-Modules, Translations of Mathematical Monographs 226, American Mathematical Society, Providence, RI, 2005. MR2125398 V.M. Manuilov and E.V. Troitsky, Hilbert $C^*$-Modules, Translations of Mathematical Monographs 226, American Mathematical Society, Providence, RI, 2005. MR2125398
J.G. Murphy, Operator Theory and $C^*$-Algebras, Academic Press, San Diego, 1990. MR1074574 0714.46041 J.G. Murphy, Operator Theory and $C^*$-Algebras, Academic Press, San Diego, 1990. MR1074574 0714.46041
G.K. Pedersen, $C^*$-Algebras and their Automorphism Groups, Academic Press, New York, 1990. MR548006 0416.46043 G.K. Pedersen, $C^*$-Algebras and their Automorphism Groups, Academic Press, New York, 1990. MR548006 0416.46043
I. Raeburn and D.P. Williams, Morita Equivalence and Continuous-Trace $C^*$-Algebras, Mathematical Surveys and Monographs. 60. Providence, RI: American Mathematical Society (AMS), 1998. MR1634408 0922.46050 I. Raeburn and D.P. Williams, Morita Equivalence and Continuous-Trace $C^*$-Algebras, Mathematical Surveys and Monographs. 60. Providence, RI: American Mathematical Society (AMS), 1998. MR1634408 0922.46050
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. MR1646207 0919.46042 10.1090/S0002-9939-99-05219-3 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. MR1646207 0919.46042 10.1090/S0002-9939-99-05219-3
N.E. Wegge-Olsen, $K$-Theory and $C^*$-Algebras - a friendly approach, Oxford University Press, Oxford, 1993. MR1222415 0780.46038 N.E. Wegge-Olsen, $K$-Theory and $C^*$-Algebras - a friendly approach, Oxford University Press, Oxford, 1993. MR1222415 0780.46038
K. Ylinen, Dual $C^*$-algebras, weakly semi-completely continuous elements, and the extreme rays of the positive cone, Ann. Acad. Sci. Fenn. Ser. A I Math. No. 599 (1975), 9 pp. MR385584 K. Ylinen, Dual $C^*$-algebras, weakly semi-completely continuous elements, and the extreme rays of the positive cone, Ann. Acad. Sci. Fenn. Ser. A I Math. No. 599 (1975), 9 pp. MR385584
K. Ylinen, Weakly completely continuous elements of C*-algebras, Proc. Amer. Math. Soc. 52 (1975), 323–326. MR383095 0312.46075 K. Ylinen, Weakly completely continuous elements of C*-algebras, Proc. Amer. Math. Soc. 52 (1975), 323–326. MR383095 0312.46075
