Free actions on C*-algebra suspensions and joins by finite cyclic groups

Abstract

We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, D ' abrowski, and Hajac. When a unital C * -algebra A admits a free action of Z/kZ, k ≥ 2, there is no equivariant map from A to the C * -algebraic join of A and the compact "quantum" group C(Z/kZ). This also resolves D ' abrowski's conjecture on unreduced suspensions of C * -algebras. Finally, we formulate a different type of noncommutative join than the previous authors, which leads to additional open problems for finite cyclic group actions.

Anticommutation in the presentations of theta-deformed spheres

“…Each R 2n ρ may be realized as a noncommutative unreduced suspension (in the sense of [9], Definition 3.4) of the unital C * -algebra C(S 2n−1 ρ ) given by the antipodal map, which is the order two homomorphism that negates each generator of the standard presentation. In general, if β generates a Z 2 action on a unital C * -algebra A, then…”

Triviality of equivariant maps in crossed products and matrix algebras

“…When we discuss continuous paths in Aut(A) or Hom (A, B), we will always mean continuous with respect to the pointwise norm topology. Both conditions assert that β is in some sense similar to the trivial action, and the conditions are motivated by examples and counterexamples from [17,16] and the previous section. In particular, condition 2 may be thought of as the demand that β is "orientation-preserving."…”

“…The conditions were determined through the computation of various examples, as in [17,16], chief among them odd-dimensional θ-deformed spheres (defined in [13,15]) and twisted versions thereof. In section 2 we consider more restrictive assumptions that guarantee nonexistence of equivariant maps from A to J(A, β).…”

Invariants in noncommutative dynamics

“…Conjecture 1.3 Type 1 has been solved in some special cases, as in [22,15]. Below we give a definition of a weak index and strong index in order to provide context for the methods used in both cases.…”

“…We begin by giving some background and notational conventions in Section 2. Next, in Section 3, we consider the invariants used to prove special cases of the Type 1 conjecture in [22,15] and describe their limitations through the construction of extremal examples. In the classical case H = C(G), G a compact torsion-free group, we produce examples which show that actions of finite local-triviality dimension (which satisfy the Type 1 conjecture by [15,Theorem 5.3]) include cases which are not distinguished by previous invariants.…”