A Noncommutative Borsuk-Ulam Theorem for Natsume-Olsen Spheres

TL;DR

This paper extends the classical Borsuk-Ulam theorem to noncommutative Natsume-Olsen spheres, showing that equivariant maps induce nontrivial K-theory maps, with implications for theta-deformed spheres and graded Banach algebras.

Contribution

It introduces a noncommutative Borsuk-Ulam theorem for Natsume-Olsen spheres, linking equivariant homomorphisms to nontrivial K-theory maps in odd dimensions.

Findings

Equivariant homomorphisms induce nontrivial K-theory maps.

The theorem applies to theta-deformed spheres of arbitrary dimension.

Results extend to graded Banach algebras.

Abstract

Natsume-Olsen noncommutative spheres are C*-algebras which generalize C(S^k) when k is odd. These algebras admit natural actions by finite cyclic groups, and if one of these actions is fixed, any equivariant homomorphism between two Natsume-Olsen 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 in terms of theta-deformed spheres of arbitrary dimension. In addition, we present theorems on graded Banach algebras, motivated by algebraic Borsuk-Ulam results of A. Taghavi.