## Abstract and Applied Analysis

### A Banach Algebraic Approach to the Borsuk-Ulam Theorem

Ali Taghavi

#### 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 ${\sum }_{i=0}^{n-1}{\lambda }^{i}f({\varphi }^{i}(x))$ must vanish at some point of ${S}^{2}$. We also discuss some noncommutative versions of the Borsuk-Ulam theorem.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 729745, 11 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495655

Digital Object Identifier
doi:10.1155/2012/729745

Mathematical Reviews number (MathSciNet)
MR2898039

Zentralblatt MATH identifier
1247.46039

#### Citation

Taghavi, Ali. A Banach Algebraic Approach to the Borsuk-Ulam Theorem. Abstr. Appl. Anal. 2012 (2012), Article ID 729745, 11 pages. doi:10.1155/2012/729745. https://projecteuclid.org/euclid.aaa/1355495655

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