A Banach Algebraic Approach to the Borsuk-Ulam Theorem. (8th March 2012) - Taghavi, Ali
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 ϕ : S 2 → S 2 be a homeomorphism of order n, and let λ ≠ 1 be an n th root of the unity, then, for every complex valued continuous function f on S 2, the function ∑ i = 0 n − 1 λ i f ( ϕ i ( x ) ) must vanish at some point of S 2 . We also discuss some noncommutative versions of the Borsuk-Ulam theorem.