Equivariant algebraic kk-theory and adjointness theorems

Abstract

Abstract. We introduce an equivariant algebraic kk-theory for G-algebras and G-graded algebras. We study some adjointness theorems related with crossed product, trivial action, induction and restriction. In particular we obtain an algebraic version of the Green-Julg Theorem which gives us a computational tool.

Graded $K$-theory and Leavitt path algebras

“…Each of these functors satisfies a universal property, which essentially says that it is universal among excisive, homotopy invariant and stable homology theories. In Section 8 we show that the adjointness theorems proved by Ellis in [12] remain valid in the hermitian setting. In particular for a subgroup H ⊂ G, the induction and restriction functors define an adjoint pair kk…”

“…We now recall the definitions of crossed products for G- * -algebras and G-graded * -algebras. We adopt the notations of [12]…”

“…In this section we introduce bivariant K-theory categories for G − Alg * ℓ and G gr − Alg * ℓ . For this purpose we first adapt the notions of polynomial homotopy invariance, extensions and excisive homology theories from [9] and [12] to the present context.…”

“…Remark 8.7. Both the algebraic analogue of the Green-Julg theorem proved in [12] and the bivariant version of Karoubi's 12-term exact sequence given in [9] can be adapted to the present context. As in the theorems above, the proofs consist in verifying that all algebras and maps involved preserve the extra structure considered.…”

“…and an elementary tensor 1 ⊗ ξ with ξ ∈ KH h 0 (ℓ) = kk h (ℓ, ℓ) to ξ (G) = ind G 1 (ξ). Next, one checks, using the explicit formula of [12,Proposition 7.4], that under the isomorphism…”