A Banach algebraic approach to the Borsuk-Ulam theorem (Q410220)

The purpose of the paper under review is to translate the classical Borsuk-Ulam theorem into the language of non-commutative geometry. The main result of the paper is the following theorem.NEWLINENEWLINEMain Theorem 1: Let \(G\) be a finite abelian group and \(A\) a \(G\)-graded Banach algebra with no non-trivial idempotents. Let \(a \in A\) be a non-trivial homogeneous element. Then \(0\) belongs to the convex hull of the spectrum \(sp(a^k)\).NEWLINENEWLINEAs a consequence, the author obtains the following corollary.NEWLINENEWLINECorollary 3.1: Let \(X\) be a compact locally path connected and simply connected space, and let \(\phi:X \to X\) be a homeomorphism of order \(n\). Assume that \( \lambda \neq 1\) is an \(n\)th root of unity. Then, for every continuous function \(f:X \to \mathbb{C}\), there exists \(x \in X\) such that \(\sum_{i=0}^{n-1} \lambda^i f(\phi^i(x))=0\).NEWLINENEWLINETaking \(X= \mathbb{S}^2\), \(\phi(x)=-x\), and \(\lambda=-1\), yields the classical Borsuk-Ulam theorem in dimension two. The author concludes by raising many interesting questions that arise naturally from his main theorem. In particular, he proposes the following conjecture.NEWLINENEWLINEA weak version of the Kaplansky-Kadison Conjecture: Let \(\Gamma\) be a torsion-free group and \(C_{red}^*(\Gamma)\) be the reduced \(C^*\) algebra of \(\Gamma\). Suppose that \(C_{red}^*(\Gamma)\) is equipped with a \(\mathbb{Z}_n\)-graded structure. Then the convex hull of the spectrum of a non-trivial homogeneous element does not contain the origin.