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Zbl 1195.46059
Arambašić, Ljiljana; Bakić, Damir; Rajić, Rajna
Finite-dimensional Hilbert $C^*$-modules. (English)
[J] Banach J. Math. Anal. 4, No. 2, 147-157, electronic only (2010). ISSN 1735-8787/e

In studying perturbations of the Wigner equation in inner product $C^*$-modules, {\it J. Chmieliński, D. Ilišević, 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. Arambašić, D. Bakić} 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 $\lim_{k\rightarrow\infty}\|v_{n_k}a-va\|=0\,\,(a\in A)$.
[Mohammad Sal Moslehian (Mashhad)]

MSC 2000:: *46L08 C*-modules
46L05 General theory of C*-algebras
46C50 Generalizations of inner products

Keywords: $C^*$-algebra; Hilbert $C^*$-module; (weakly) compact operator; finite-dimensional $C^*$-algebra; finite-dimensional Hilbert $C^*$-module

Citations: Zbl 1153.81342; Zbl 1188.46036

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