Finite-dimensional Hilbert \(C^*\)-modules
In studying perturbations of the Wigner equation in inner product \(C^*\)-modules, \textit{J. Chmieliński, D. Ilišević, M. S. Moslehian} and \textit{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, \textit{Lj. Arambašić, D. Bakić} and \textit{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 \(\lim_{k\rightarrow\infty}\|v_{n_k}a-va\|=0\,\,(a\in A)\).
finite-dimensional Hilbert \(C^*\)-module
scientific article; zbMATH DE number 5702421