We investigate the problem of identifying isolated points in spaces of continuous homomorphisms from ordered AL-algebras to normed algebras. Building on recent work of Bobrowski and the author [Dissertationes Math. (Rozprawy Mat.) 577 (2022), 1–69], we consider homomorphisms from semigroup algebras associated with non-locally-null, Haar measurable subsemigroups of locally compact groups. A key novelty is the removal of the assumption that the semigroup algebra admits an approximate identity of bound 1, which was essential in prior work. Our main result shows that for any continuous character <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> on a suitable semigroup S, the homomorphism <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/233_2026_10654_IEq3_HTML.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="120" Type="Linedraw" Width="282" /> </InlineMediaObject> </InlineEquation> is isolated in the space of all continuous homomorphisms from <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^1(S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to a unital normed algebra <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathfrak {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">A</mi> </math></EquationSource> </InlineEquation>. Here <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^1(S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denotes the semigroup algebra of S, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$e_{\mathfrak {A}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>e</mi> <mi mathvariant="fraktur">A</mi> </msub> </math></EquationSource> </InlineEquation> is the unit of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathfrak {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">A</mi> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/233_2026_10654_IEq9_HTML.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="120" Type="Linedraw" Width="25" /> </InlineMediaObject> </InlineEquation> is the character on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L^1(S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> determined by <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>. This follows from a quantitative rigidity theorem: any continuous homomorphism within operator-norm distance less than 1 from <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/233_2026_10654_IEq12_HTML.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="120" Type="Linedraw" Width="64" /> </InlineMediaObject> </InlineEquation> must coincide with it. Along the way, we develop a general framework for constructing and analysing such homomorphisms, including integral and associated representations, which serve not only to establish the main result, but also to characterise characters on semigroup algebras.

On homomorphisms from semigroup algebras: generation, characters, and isolated points

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2026年7月7日

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Wojciech Chojnacki

摘要

We investigate the problem of identifying isolated points in spaces of continuous homomorphisms from ordered AL-algebras to normed algebras. Building on recent work of Bobrowski and the author [Dissertationes Math. (Rozprawy Mat.) 577 (2022), 1–69], we consider homomorphisms from semigroup algebras associated with non-locally-null, Haar measurable subsemigroups of locally compact groups. A key novelty is the removal of the assumption that the semigroup algebra admits an approximate identity of bound 1, which was essential in prior work. Our main result shows that for any continuous character $\sigma $ $σ$ on a suitable semigroup S, the homomorphism is isolated in the space of all continuous homomorphisms from $L^1(S)$ $L^{1} (S)$ to a unital normed algebra $\mathfrak {A}$ $A$ . Here $L^1(S)$ $L^{1} (S)$ denotes the semigroup algebra of S, $e_{\mathfrak {A}}$ $e_{A}$ is the unit of $\mathfrak {A}$ $A$ , and is the character on $L^1(S)$ $L^{1} (S)$ determined by $\sigma $ $σ$ . This follows from a quantitative rigidity theorem: any continuous homomorphism within operator-norm distance less than 1 from must coincide with it. Along the way, we develop a general framework for constructing and analysing such homomorphisms, including integral and associated representations, which serve not only to establish the main result, but also to characterise characters on semigroup algebras.