Finitely-Generated Projective Modules over the θ-Deformed 4-Sphere

Abstract

Abstract. We investigate the "θ-deformed spheres" C(S 3 θ ) and C(S 4 θ ), where θ is any real number. We show that all finitelygenerated projective modules over C(S 3 θ ) are free, and that C(S 4 θ ) has the cancellation property. We classify and construct all finitelygenerated projective modules over C(S 4 θ ) up to isomorphism. An interesting feature is that if θ is irrational then there are nontrivial "rank-1" modules over C(S 4 θ ). In that case, every finitelygenerated projective module over C(S 4 θ ) is a sum of a rank-1 module and a free module. If θ is rational, the situation mirrors that for the commutative case θ = 0.

A noncommutative Borsuk-Ulam theorem for Natsume-Olsen spheres

“…Natsume and Olsen chose to specify K 1 ∼ = Z more concretely in terms of a noncommutative Toeplitz algebra, and we adopt this identification: an invertible matrix M over C(S 2n−1 ρ ) is indentified with the negative index of its Toeplitz operator. Next, the θ-deformed even spheres C(S 2m ρ ) may be be found in [4], and they are described via generators and relations in [17] (among other places) with some results on projective modules. These spheres also admit a natural antipodal action, giving us a corollary (Corollary 4.6) of the above result.…”

Towards a noncommutative Brouwer fixed-point theorem

“…• γ is the identity on A. ⋄ We mention that the n-dimensional quantum θ-balls have been defined for n even in [20] and they are just isomorphic to ΓS n−1 θ . Thus Conjecture 3.2, Proposition 3.3 and Conjecture 3.4 hold true for these families.…”

Projective Modules Over Quantum Projective Line

“…Going beyond the K-theoretic study of C*-algebras that classifies finitely generated projective modules only up to stable isomorphism, some successes have been achieved in the study of cancellation problem, made popular by Rieffel [16,17], that classifies finitely generated projective modules up to isomorphism, for some quantum algebras [17,18,20,1,14]. It is of interest and a natural question to classify finitely generated projective modules over C (P n (T )) and identify the line bundles L k among them, beside finding the K-groups of C (P n (T )).…”