### Compact and “compact” operators on standard Hilbert modules over $C^*$-algebras

Zlatko Lazović

#### Abstract

We construct a topology on the standard Hilbert module $H_{\mathcal{A}}$ over a unital $C^*$-algebra and topology on $H_ \mathcal {A} ^{＃}$ (the extension of the module $H_{\mathcal{A}}$ by the algebra $\mathcal{A}^{**}$) such that any “compact” operator (i.e. any operator in the norm closure of the linear span of the operators of the form $z\mapsto x \langle y,z \rangle$, $x,y\in H_{\mathcal{A}}$ (or $x,y \in H_ \mathcal {A} ^{＃}$)) maps bounded sets into totally bounded sets.

#### Article information

Source
Adv. Oper. Theory, Volume 3, Number 4 (2018), 829-836.

Dates
Accepted: 29 June 2018
First available in Project Euclid: 7 July 2018

https://projecteuclid.org/euclid.aot/1530928843

Digital Object Identifier
doi:10.15352/aot.1806-1382

Subjects
Primary: 46L08: $C^*$-modules
Secondary: 47B07: Operators defined by compactness properties

#### Citation

Lazović, Zlatko. Compact and “compact” operators on standard Hilbert modules over $C^*$-algebras. Adv. Oper. Theory 3 (2018), no. 4, 829--836. doi:10.15352/aot.1806-1382. https://projecteuclid.org/euclid.aot/1530928843

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