A Banach Algebraic Approach to the Borsuk-Ulam Theorem

Ali Taghavi

Abstract and Applied Analysis, 2012, vol. 2012, 1-11

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 𠜙 ∶ 𠑆 2 → 𠑆 2 be a homeomorphism of order 𠑛 , and let 𠜆 ≠1 be an 𠑛 th root of the unity, then, for every complex valued continuous function 𠑓 on 𠑆 2 , the function ∑ 𠑛 − 1 𠑖 = 0 𠜆 𠑖 𠑓 ( 𠜙 𠑖 ( 𠑥 ) ) must vanish at some point of 𠑆 2 . We also discuss some noncommutative versions of the Borsuk-Ulam theorem.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:729745

DOI: 10.1155/2012/729745

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