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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2306.15979 |
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| _version_ | 1866910588530065408 |
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| author | Meyerovitch, Tom Solan, Omri Nisan |
| author_facet | Meyerovitch, Tom Solan, Omri Nisan |
| contents | Classical theorems from the early 20th century state that any Haar measurable homomorphism between locally compact groups is continuous. In particular, any Lebesgue-measurable homomorphism $ϕ:\mathbb{R} \to \mathbb{R}$ is of the form $ϕ(x)=ax$ for some $a \in \mathbb{R}$. In this short note, we prove that any Lebesgue measurable function $ϕ:\mathbb{R} \to \mathbb{R}$ that vanishes under any $d+1$ ``difference operators'' is a polynomial of degree at most $d$. More generally, we prove the continuity of any Haar measurable polynomial map between locally compact groups, in the sense of Leibman. We deduce the above result as a direct consequence of a theorem about the automatic continuity of cocycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15979 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Automatic continuity of Polynomial maps and cocycles Meyerovitch, Tom Solan, Omri Nisan Geometric Topology 22D50, 28C10 Classical theorems from the early 20th century state that any Haar measurable homomorphism between locally compact groups is continuous. In particular, any Lebesgue-measurable homomorphism $ϕ:\mathbb{R} \to \mathbb{R}$ is of the form $ϕ(x)=ax$ for some $a \in \mathbb{R}$. In this short note, we prove that any Lebesgue measurable function $ϕ:\mathbb{R} \to \mathbb{R}$ that vanishes under any $d+1$ ``difference operators'' is a polynomial of degree at most $d$. More generally, we prove the continuity of any Haar measurable polynomial map between locally compact groups, in the sense of Leibman. We deduce the above result as a direct consequence of a theorem about the automatic continuity of cocycles. |
| title | Automatic continuity of Polynomial maps and cocycles |
| topic | Geometric Topology 22D50, 28C10 |
| url | https://arxiv.org/abs/2306.15979 |