A Banach algebraic approach to the Borsuk-Ulam theorem

DOI10.1155/2012/729745zbMATH Open1247.46039arXiv1110.0091OpenAlexW3102653715WikidataQ58696674 ScholiaQ58696674MaRDI QIDQ410220

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Publication date: 3 April 2012

Published in: (Search for Journal in Brave)

Abstract: Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two dimensional Borsuk-Ulam theorem as follows: Let

p h i : S^{2} i g h t a r r o w S^{2}

be a homeomorphism of order n and

l a m b d a e q 1

be an nth root of the unity, then for every complex valued continuous function

f

on

S^{2}

the function

s u m_{i = 0}^{n - 1} l a m b d a^{i} f (p h i^{i} (x))

must be vanished at some point of

S^{2}

. We give a generalization in term of action of compact groups. We also discuss about some noncommutative versions of the Borsuk- Ulam theorem

Full work available at URL: https://arxiv.org/abs/1110.0091

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