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Finite-dimensional Hilbert ${C}^{*}$-modules. (English) Zbl 1195.46059

In studying perturbations of the Wigner equation in inner product ${C}^{*}$-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 $\left({v}_{n}\right)$ in a Hilbert ${C}^{*}$-module $V$, there are a subsequence $\left({v}_{{n}_{k}}\right)$ of $\left({v}_{n}\right)$ and $v\in V$ such that, for every $y\in V$, ${lim}_{k\to \infty }\parallel 〈y,{v}_{{n}_{k}}〉-〈y,v〉\parallel =0$. They proved that condition [H] is satisfied in every Hilbert ${C}^{*}$-module over a finite-dimensional ${C}^{*}$-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 $A$-module satisfies condition [H], then $A$ must be finite-dimensional.

In the paper under review, the authors characterize the finite-dimensional Hilbert ${C}^{*}$-modules in terms of the convergence of certain sequences. More precisely, they prove that, if $V$ is a full right Hilbert module over a ${C}^{*}$-algebra $A$, then the following statements are mutually equivalent: (i) $V$ is finite-dimensional; (ii) $A$ and the ${C}^{*}$-algebra $K\left(V\right)$ of compact operators on $V$ are finite-dimensional; (iii) for every bounded sequence $\left({v}_{n}\right)$ in $V$, there are a subsequence $\left({v}_{{n}_{k}}\right)$ of $\left({v}_{n}\right)$ and $v\in V$ such that ${lim}_{k\to \infty }\parallel {v}_{{n}_{k}}a-va\parallel =0$ $\left(a\in A\right)$ and ${lim}_{k\to \infty }\parallel 〈y,{v}_{{n}_{k}}〉-〈y,v〉\parallel =0$ $\left(y\in V\right)$; (iv) $K\left(V\right)$ is a unital ${C}^{*}$-algebra, and for every bounded sequence $\left({v}_{n}\right)$ in $V$, there are a subsequence $\left({v}_{{n}_{k}}\right)$ of $\left({v}_{n}\right)$ and $v\in V$ such that ${lim}_{k\to \infty }\parallel 〈y,{v}_{{n}_{k}}〉-〈y,v〉\parallel =0$ $\left(y\in V\right)$; (v) $A$ is a unital ${C}^{*}$-algebra, and for every bounded sequence $\left({v}_{n}\right)$ in $V$, there are a subsequence $\left({v}_{{n}_{k}}\right)$ of $\left({v}_{n}\right)$ and $v\in V$ such that ${lim}_{k\to \infty }\parallel {v}_{{n}_{k}}a-va\parallel =0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left(a\in A\right)$.

##### MSC:
 46L08 ${C}^{*}$-modules 46L05 General theory of ${C}^{*}$-algebras 46C50 Generalizations of inner products