Tensor category equivariant KK-theory (Q6608021)

This article develops an equivariant version of Kasparov's bivariant \(K\)-theory for \(C^\ast\)-algebras equipped with an action of a \(C^\ast\)-tensor category. This generalises actions of compact quantum groups. This, in turn, generalises actions of both compact and discrete groups as special cases, using also Takesaki-Takai duality. The article first sets up the theory, including the usual constructions like the Kasparov product. Then it establishes a Cuntz quasihomomorphism picture for it, which is then used to characterise the equivariant KK-theory through a universal property. The triangulated category structure is also explained. The latter is useful to generalise some aspects of the Baum-Connes assembly map. Actually, the universal property has three versions, which deal with functors defined on categories that have equivariant *-homomorphisms, cocycle *-homomorphisms, or isomorphism classes of equivariant Hilbert bimodules as arrows. It is worth pointing out that Hilbert bimodules in this article only carry one and not two inner products; so they are what are also called \(C^\ast\)-correspondences in some recent literature. To prove the universal property, an equivariant variant of free products is introduced and shown to behave in certain ways like usual free products.\N\NAn important motivation of the article is the fact that two monoidally equivalent compact quantum groups have equivalent KK-theories. This suggests that the equivariant KK-theory for a compact quantum group should be definable using only its representation category. The construction in this article shows that this is indeed the case. Even more, it is shown that two \(C^\ast\)-tensor categories have equivalent KK-theories once they are weakly Morita equivalent. This implies a partial crossed product version of Takesaki-Takai duality, where the crossed product is not taken with respect to the whole group, but only with respect to a finite abelian normal subgroup of it.\N\NThe article pays special attention to 3-cocycle twists of discrete groups. These are \(C^\ast\)-tensor categories whose actions are anomalous twisted group actions, that is, the usual cocycle condition for a twisted action is replaced by asking a specified 3-cocycle to come up. It is worth mentioning that such anomalous actions have been studied in the context of T-duality because they allow to represent certain missing T-duals, see [\textit{P. Bouwknegt} et al., Commun. Math. Phys. 264, No. 1, 41--69 (2006; Zbl 1115.46063)] and [\textit{R. Meyer} and \textit{U. Pennig}, Commun. Math. Phys. 346, No. 1, 115--142 (2016; Zbl 1361.46053)]. Under an extra cohomological condition, it turns out that the equivariant KK-theory for such anomalous actions is equivalent to the usual KK-theory in an interesting way. This allows to carry over important tools such as the Dirac--dual Dirac method for proving the Baum-Connes conjecture.\N\NThe article highlights categorical aspects and explains some constructions in careful detail. Many basic constructions and the relevant equivariant version of Kasparov's technical theorem are discussed in appendices. I noticed an imprecise formulation in the paragraph above Propostion 4.9. The relevant convex set consists only of the locally compact perturbations that make \((E,F)\) special, not all operators~\(F\) that make \((E,F)\) special.