In studying perturbations of the Wigner equation in inner product -modules, J. Chmieliński, D. Ilišević, M. S. Moslehian and Gh. Sadeghi [J. Math. Phys. 49, No. 3, 033519, 8 p. (2008; Zbl 1153.81342)] introduced the condition [H] stating that, for every bounded sequence in a Hilbert -module , there are a subsequence of and such that, for every , . They proved that condition [H] is satisfied in every Hilbert -module over a finite-dimensional -algebra. Later, Lj. Arambašić, D. Bakić and M. S. Moslehian [Oper. Matrices 3, No. 2, Article ID 14, 235–240 (2009; Zbl 1188.46036)] proved that, if a full Hilbert -module satisfies condition [H], then must be finite-dimensional.
In the paper under review, the authors characterize the finite-dimensional Hilbert -modules in terms of the convergence of certain sequences. More precisely, they prove that, if is a full right Hilbert module over a -algebra , then the following statements are mutually equivalent: (i) is finite-dimensional; (ii) and the -algebra of compact operators on are finite-dimensional; (iii) for every bounded sequence in , there are a subsequence of and such that and ; (iv) is a unital -algebra, and for every bounded sequence in , there are a subsequence of and such that ; (v) is a unital -algebra, and for every bounded sequence in , there are a subsequence of and such that .