On closed range multipliers on topological algebras (Q2740972)
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scientific article; zbMATH DE number 1642185
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On closed range multipliers on topological algebras |
scientific article; zbMATH DE number 1642185 |
Statements
28 October 2001
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multiplier
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ascent
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descent
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Fréchet locally convex algebra
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locally \(C^*\)-algebra
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semiprime
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Fréchet algebra without order
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On closed range multipliers on topological algebras (English)
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The authors obtain some equivalent conditions for continuous linear operators on a Fréchet space to have closed range. They apply these conditions to a multiplier \(T\) on a Fréchet algebra \(A\) without order and they prove, among other conditions, that \(T\) has a \(g\)-inverse \(S\) such that \(TS=ST\) if and only if \(TA\oplus \operatorname {Ker}T=A\) if and only \(T=PB=BP\), where \(P,B\) are multipliers, \(B\) is invertible and \(P\) is an idempotent. Furthermore, it is shown that if \(A\) is a Fréchet locally \(C^*\)-algebra and \(T\) a multiplier of \(A\), then the range of \(T\) is closed if and only if \(T^2A=TA\).
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