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 let $\lambda \ne 1$ be an $n$th root of the unity, then, for every complex valued continuous function $f$ on $S^2$, the function ${\sum }_{i=0}^{n-1}{\lambda }^{i}f\left({\varphi }^i\right(x\left)\right)$ must vanish at some point of $S^2$. We also discuss some noncommutative versions of the Borsuk-Ulam theorem.