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Abstract and Applied Analysis
Volume 2012, Article ID 729745, 11 pages
http://dx.doi.org/10.1155/2012/729745
Research Article

A Banach Algebraic Approach to the Borsuk-Ulam Theorem

Faculty of Mathematics and Computer Science, Damghan University, Damghan, Iran

Received 1 October 2011; Accepted 12 December 2011

Academic Editor: David Perez-Garcia

Copyright © 2012 Ali Taghavi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 πœ™ ∢ 𝑆 2 β†’ 𝑆 2 be a homeomorphism of order 𝑛 , and let πœ† β‰  1 be an 𝑛 th root of the unity, then, for every complex valued continuous function 𝑓 on 𝑆 2 , the function βˆ‘ 𝑛 βˆ’ 1 𝑖 = 0 πœ† 𝑖 𝑓 ( πœ™ 𝑖 ( π‘₯ ) ) must vanish at some point of 𝑆 2 . We also discuss some noncommutative versions of the Borsuk-Ulam theorem.