Measures of noncompactness on the standard Hilbert $C^*$-module
Abstract
We define a measure of noncompactness $\lambda$ on the standard Hilbert $C^*$-module $l^2(\mathcal A)$ over a unital $C^*$-algebra, such that $\lambda(E)=0$ if and only if $E$ is $\mathcal A$-precompact (i.e.\ it is $\varepsilon$-close to a finitely generated projective submodule for any $\varepsilon>0$) and derive its properties. Further, we consider the known, Kuratowski, Hausdorff and Istrăţescu measure of noncomapctnes on $l^2(\mathcal A)$ regarded as a locally convex space with respect to a suitable topology, and obtain their properties as well as some relationship between them and introduced measure of noncompactness $\lambda$.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2017
- DOI:
- 10.48550/arXiv.1711.09466
- arXiv:
- arXiv:1711.09466
- Bibcode:
- 2017arXiv171109466K
- Keywords:
-
- Mathematics - Operator Algebras;
- Primary: 46L08;
- Secondary: 47H08;
- 54E15
- E-Print:
- 15 pages