A Banach Algebraic Approach to the Borsuk-Ulam Theorem

الملخص EN

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:S2→S2 be a homeomorphism of order n, and let λ≠1 be an nth root of the unity, then, for every complex valued continuous function f on S2, the function ∑i=0n−1λif(ϕi(x)) must vanish at some point of S2.

We also discuss some noncommutative versions of the Borsuk-Ulam theorem.