@article{1195.46059,
author="Aramba\v si\'c, Ljiljana and Baki\'c, Damir and Raji\'c, Rajna",
title="{Finite-dimensional Hilbert $C^*$-modules.}",
language="English",
journal="Banach J. Math. Anal. ",
volume="4",
number="2",
pages="147-157",
year="2010",
abstract="{In studying perturbations of the Wigner equation in inner product
    $C^*$-modules, {\it J. Chmieli\'nski, D. Ili\v sevi\'c, M. S. Moslehian} and
    {\it 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 $(v_n)$ in a Hilbert $C^*$-module $V$, there are a subsequence
    $(v_{n_k})$ of $(v_n)$ and $v\in V$ such that, for every $y\in V$,
    $\lim_{k\rightarrow\infty}\|\langle y,v_{n_k}\rangle -\langle
    y,v\rangle\|=0$. They proved that condition [H] is satisfied in every
    Hilbert $C^*$-module over a finite-dimensional $C^*$-algebra. Later, {\it
    Lj. Aramba\v si\'c, D. Baki\' c} and {\it 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 $K(V)$ of compact operators on $V$ are finite-dimensional;
    (iii) for every bounded sequence $(v_n)$ in $V$, there are a subsequence
    $(v_{n_k})$ of $(v_n)$ and $v\in V$ such that
    $\lim_{k\rightarrow\infty}\|v_{n_k}a-va\|=0$ $(a\in A)$ and
    $\lim_{k\rightarrow\infty}\|\langle y,v_{n_k}\rangle -\langle
    y,v\rangle\|=0$ $(y\in V)$; (iv) $K(V)$ is a unital $C^*$-algebra, and for
    every bounded sequence $(v_n)$ in $V$, there are a subsequence $(v_{n_k})$
    of $(v_n)$ and $v\in V$ such that $\lim_{k\rightarrow\infty}\|\langle
    y,v_{n_k}\rangle -\langle y,v\rangle\|=0$ $(y\in V)$; (v) $A$ is a unital
    $C^*$-algebra, and for every bounded sequence $(v_n)$ in $V$, there are a
    subsequence $(v_{n_k})$ of $(v_n)$ and $v\in V$ such that
    $\lim_{k\rightarrow\infty}\|v_{n_k}a-va\|=0\,\,(a\in A)$.}",
reviewer="{Mohammad Sal Moslehian (Mashhad)}",
keywords="{$C^*$-algebra; Hilbert $C^*$-module; (weakly) compact operator;
    finite-dimensional $C^*$-algebra; finite-dimensional Hilbert $C^*$-module}",
classmath="{*46L08 (C*-modules)
46L05 (General theory of C*-algebras)
46C50 (Generalizations of inner products)
}",
}