A Banach algebraic Approach to the Borsuk-Ulam Theorem

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 $\varphi :S^2\to S^2$ be a homeomorphism of order n and $\lambda \ne 1$ be an nth root of the unity, then for every complex valued continuous function $f$ on $S^2$ the function ${\sum }_{i=0}^{n-1}{\lambda }^{i}f({\varphi }^i(x))$ 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

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