Advances in Operator Theory

Compact and ``compact'' operators on standard Hilbert modules over $C^*$-algebras

Zlatko Lazović

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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 $z\mapsto x \lange y,z \rangle$, $x,y\in H_{\mathcal{A}}$ (or $x,y\in H_{\mathcal{A}}^{#}$)) maps bounded sets into totally bounded sets.

Article information

Adv. Oper. Theory, Advance publication (2018), 8 pages.

Received: 12 June 2018
Accepted: 29 June 2018
First available in Project Euclid: 7 July 2018

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Digital Object Identifier

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

Hilbert module ‎compact operator‎ ‎ locally convex topology


Lazović, Zlatko. Compact and ``compact'' operators on standard Hilbert modules over $C^*$-algebras. Adv. Oper. Theory, advance publication, 7 July 2018. doi:10.15352/aot.1806-1382.

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