Topological realizations of Calkin algebras on Fréchet domains of unbounded operator algebras
type: Item
instance of: scholarly article
title: Topological realizations of Calkin algebras on Fréchet domains of unbounded operator algebras
zbMATH Open document ID: 0632.47036
DOI: 10.4171/ZAA/217
author: Konrad Schmüdgen
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1986-01-01T00:00:00Z
review text: Let D be a dense linear subspace of a complex Hilbert space H, and let \(L^+(D)\) be the *-algebra of all linear transformations a on H such that \(D\subset D(a^*)\) and \(a^*D\subset D\). An Op*-algebra on D is a *-subalgebra of \(L^+(D)\). The graph topology from \(L^+(D)\) on D induces a uniform topology on \(L^+(D)\). Let C(D) denote the *-ideal of all operators in \(L^+(D)\) which map each bounded subset of D into a relatively compact subset of D. Then the Calkin algebra \(L^+(D)/C(D)\) with the quotient topology is considered. For each free ultrafilter of \({\mathbb{N}}^ a \)faithful *-representation of \(L^+(D)/C(D)\) with continuous inverse is constructed such that the image is an Op*-algebra. The continuity of this representation is studied, and in connection with former results of the author's [Rep. Math. Phys. 17, 359-371 (1980; Zbl 0475.47030)], where D is subject to some additional assumptions, necessary and sufficient continuity conditions are proved.
Mathematics Subject Classification ID: 47L60
Mathematics Subject Classification ID: 47L30
zbMATH DE Number: 4028227
zbMATH Keywords: Op*-algebra
zbMATH Keywords: graph topology
zbMATH Keywords: *-ideal
zbMATH Keywords: Calkin algebra
zbMATH Keywords: free ultrafilter
zbMATH Keywords: representation
reviewed by: Gerhard Garske
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/217
OpenAlex ID: W2566791838
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description: scientific article; zbMATH DE number 4028227