A descent homomorphism for semimultiplicative sets

Burgstaller, Bernhard

doi:10.48550/arxiv.1111.4160

Cited by 2 publications

(10 citation statements)

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“…for s ∈ S, e ∈ E and ξ ∈ E, and so U (1) (1 e ξ) ⊗ U (2) (η) = U (1) (ξ) ⊗ U (2) (1 e η) in E 1 ⊗ E 2 for η ∈ E 2 . Both the exterior and internal tensor product are compatible Hilbert modules.…”

Section: Compatible S-equivariant Kk-theorymentioning

confidence: 99%

“…For crossed products of inverse semigroups we will use constructions from three sources: Sieben's full crossed product [12], Khoshkam and Skandalis' reduced and full crossed products [6], and the author's full crossed product [2]. In any case, an S-action is an inverse semigroup homomorphism from S to some objects related to the C * -algebra.…”

Section: Crossed Productsmentioning

confidence: 99%

“…In any case, an S-action is an inverse semigroup homomorphism from S to some objects related to the C * -algebra. Khoshkam and Skandalis [6] are most general and allow morphisms from S into the isomorphisms between quotients of ideals, Sieben [12] into partial automorphisms, and the author [2] into endomorphisms which are also partial isometries (see Section 4 of [2]), the smallest class.…”

Section: Crossed Productsmentioning

confidence: 99%

“…If s ∈ S and x ∈ X then we write sx for the unique single element g ∈ s ⊆ G such that s(g) = x. Analogously we define xs.We use inverse semigroup equivariant KK-theory KK S as introduced in[3] by regarding S as a semimultiplicative set. The definition of KK S is also summarized in Section 3 of[2], and we prefer using it as our reference.The evaluation of an S-action α on an S-Hilbert C * -algebra A ([2, Def.3.1]) is usually denoted by s(a) rather than α s (a) (s ∈ S and a ∈ A). The restriction to E is an action α| E : E −→ Z(L(A)) into the center of the multiplier algebra of A, see last section of [3] or [2, Lemma 5.10].…”

mentioning

confidence: 99%

“…If α is a compatible S-action then β = (β s : A s * s → A ss * ) s∈S , β s := α s | A s * s , defines a partial action in Sieben's sense on A which is compatible with the C 0 (X)-structure. In the other direction, given an action β, α s (a) := β s (1 s * s • a) (s ∈ S and a ∈ A) defines a compatible S-action on A.The author's full crossed product from[2] coincides with the one of Khoshkam-Skandalis by[2, Corollary 8.4], and is defined as follows. The crossed product A ⋊ S is the enveloping C * -algebra of the Banach * -algebra ℓ 1 (S, A).…”

mentioning

confidence: 99%

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Equivariant $KK$-theory of $r$-discrete groupoids and inverse semigroups

Burgstaller¹

2012

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For an r-discrete Hausdorff groupoid G and an inverse semigroup S of slices of G there is an isomorphism between G-equivariant KK-theory and compatible S-equivariant KK-theory. We use it to define descent homomorphisms for S, and indicate a Baum-Connes map for inverse semigroups. Also findings by Khoshkam and Skandalis for crossed products by inverse semigroups are reflected in KK-theory.to-one correspondence between a G-action on A and an S-action on A (in Sieben's sense [12]). Moreover, they found an isomorphism between the corresponding groupoid crossed product A ⋊ G and the corresponding inverse semigroup crossed product A ⋊S (in Sieben's sense [12]). Khoshkam and Skandalis [6] introduced another crossed product A ⋊ S for S which usually significantly differs from Sieben's crossed product, but have shown in [6] that (A ⋊ E) ⋊S and A ⋊ S are isomorphic, where E denotes the set of idempotent elements of S.In this note we aim to extend these results to a KK-theoretical level. We use semimultiplicative set equivariant KK-theory KK S introduced in [3], but also slightly adapt it such that the underlying C 0 (X)-structure is compatible with the module multiplication and denote this so-called compatible equivariant KK-theory by KK S .

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