A Banach Algebraic Approach to the Borsuk-Ulam Theorem
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چکیده
and Applied Analysis 3 Proof. This can be proved by induction on order ofG as follows. The first step of the induction is obvious since the only group of order 2 is Z2. Let G be a finite Abelian group and g ∈ G where g / e. If G is not a cyclic group, then there is a subgroup H which does not contain g, so a is a nontrivial homogenous element of the induced G/H-graded structure for A. Now the order of G/H is strictly less than the order of G. So an induction argument on order of G completes the proof. Note that the existence of a Zn-graded structure for a Banach algebra A is equivalent to existence of a bounded multiplicative operator T : A → A with T Id. For any such operator, we choose a root of unity λ/ 1 and observe that the decomposition
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