On the \(\ast\)-semisimplicity of the \(\ell^1\)-algebra on an abelian \(\ast\)-semigroup (Q2872018)
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scientific article; zbMATH DE number 6245026
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the \(\ast\)-semisimplicity of the \(\ell^1\)-algebra on an abelian \(\ast\)-semigroup |
scientific article; zbMATH DE number 6245026 |
Statements
14 January 2014
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semigroup algebra
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semisimplicity
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semicharacter
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\(\ast\)-semigroup
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Banach \(\ast\)-algebra
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\(\ast\)-semisimplicity
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On the \(\ast\)-semisimplicity of the \(\ell^1\)-algebra on an abelian \(\ast\)-semigroup (English)
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If \(S\) is an abelian semigroup, Hewitt and Zuckerman showed that the semisimplicity of the semigroup algebra \(\ell^1(S)\) is equivalent to each of the following: i) bounded semicharacters of \(S\) separate points of \(S\); ii) property \(P_0: s,t\in S\) and \(s^2=t^2=st\) imply \(s=t\). When \(S\) is a \(\ast\)-semigroup, the authors prove analogous results for \(\ast\)-semisimplicity of \(\ell^1(S)\). They also identify several analogues of the property \(P_0\) that are not the `right ones' in this context.
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