# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2015apr21.2071

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

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