A Green--Julg isomorphism for inverse semigroups

Abstract

For every finite unital inverse semigroup S and S-C * -algebra A we establish an isomorphism between KK S (C, A) and K(A ⋊ S). This extends the classical Green-Julg isomorphism from finite groups to finite inverse semigroups.

Attempts to define a Baum--Connes map via localization of categories for inverse semigroups

“…Also recall that Res pG G (A) = pA. For the second isomorphism we decomposeB ∼ = pB ⊕ (1 − p)B and note that KK G (pA, (1 − p)B) = 0 since p(a)ξ(1 − p)(b) = 0 for a ∈ A, ξ ∈ E and b ∈ B, where (E, T ) is a cycle.The next lemma is similar to the fact that the K-theory group KK(C, B) = K(B) is countable. It is immediately evidently true in IK-theory by the Green-Julg isomorphism IK H (C, A) ∼ = K(A ⋊ H) in[2]. For all compact subinverse semigroups H ⊆ G KK H (Res H G C, B) is countable for all B ∈ KK G and commutes with countable direct sums in the variable B.…”