  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.
