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Zbl 1188.46036
Arambašić, Ljiljana; Bakić, Damir; Moslehian, Mohammad Sal
A characterization of Hilbert $C^{\ast}$-modules over finite dimensional $C^{\ast}$-algebras. (English)
[J] Oper. Matrices 3, No. 2, Article ID 14, 235-240 (2009). ISSN 1846-3886

Hilbert $C^*$-modules are generalizations of Hilbert spaces, but the theory of Hilbert $C^*$-modules is different from the theory of Hilbert spaces; for example, not all closed submodules of a given Hilbert $C^*$-modules are orthogonally complemented. It is well-known that the closed unit ball of a Hilbert space $H$ is weakly sequentially compact. This result is not true for Hilbert $C^*$-modules. A sequence $\{\xi_n\}_n$ in a Hilbert $C^*$-module $E$ over a $C^*$-algebra $A$ is weakly convergent to an element $\xi\in E$ if the sequence $\{\langle\xi_n,\eta\rangle\}_n$ converges to $\langle\xi,\eta\rangle$ with respect to the $C^*$-norm on $A$, for each $\eta\in E$. The authors prove that the closed unit ball of a full Hilbert $C^*$-module $E$ over a $C^*$-algebra $A$ is weakly sequentially compact if and only if the $C^*$-algebra $A$ is finite-dimensional.
[Maria Joiţa (Bucureşti)]

MSC 2000:: *46L08 C*-modules
46L05 General theory of C*-algebras
46L10 General theory of von Neumann algebras

Keywords: Hilbert $C^{\ast}$-module; finite-dimensional $C^{\ast}$-algebra; $C^{\ast}$-algebra of compact operators

Cited in: Zbl 1195.46059

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