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Journal of Operator Theory

Volume 75, Issue 2, Spring 2016  pp. 337-366.

A noncommutative Borsuk-Ulam theorem for Natsume-Olsen spheres

Authors:  Benjamin W. Passer
Author institution: Mathematics Department, Washington University in St. Louis, St. Louis, MO, 63130, U.S.A.

Summary: The odd $\theta$-deformed spheres are $C^*$-algebras that admit natural actions by finite cyclic groups, and if one of these actions is fixed, any equivariant homomorphism between two spheres of the same dimension induces a nontrivial map on odd $K$-theory. This result is an extended, noncommutative Borsuk-Ulam theorem in odd dimension, and just as in the topological case, this theorem has many (almost) equivalent formulations for $\theta$-deformed spheres of arbitrary dimension. We also present theorems on graded Banach algebras, motivated by algebraic Borsuk-Ulam results of A. Taghavi.

Keywords: $C^*$-algebra, noncommutative, sphere, Borsuk-Ulam, $K$-theory, group action, deformation

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