A descent homomorphism for semimultiplicative sets

Abstract

We define and provide some basic analysis of various types of crossed products by semimultiplicative sets, and then prove a KK-theoretical descent homomorphisms for semimultiplicative sets in accord with the descent homomorphism for discrete groups.definition of equivariant KK-theory for inversely generated semigroups. In Section 9 we compare semimultiplicative set G-equivariant KK-theory with Kasparov's G-equivariant KK-theory when G is a group. Sections 11-13 occupy the proof of the descent homomorphism, which is an adaption of Kasparov's proof in [9].DESCENT HOMOMORPHISM 3 2. Semimultiplicative sets Definition 2.1. A (general) semimultiplicative set G is a set endowed with a subset G (2) ⊆ G × G and a map (written as a multiplication)satisfying the following weak associativity condition: s(tu) = (st)u whenever both expressions are defined (s, t, u ∈ G).There is a similar notion called a semigroupoid ([7]). A semigroupoid is an associative semimultiplicative set with the property that (st)u is defined if and only if st and tu is defined. For instance, groupoids and small categories are semigroupoids. In general, however, an associative semimultiplicative set is not a semigroupoid, a typical example being a ring R without the zero element, so the semimultiplicative set G = R\{0} under the multiplication inherited from R. Examples for associative semimultiplicative sets include groups, groupoids, small categories, inverse semigroups, semigroups, semigroupoids. An associative semimultiplicative set is also called a partial semigroup in the literature (see [2]).We remark that the weak associativity condition for a general semimultiplicative set is not essential in this paper. A general semimultiplicative set is always realized by associative actions, so we require the weak associativity without essential loss of generality. However, for instance, an arbitrary subset of a group is a general but not necessarily an associative semimultiplicative set. Now the point is that general and associative semimultiplicative sets G yield different classes of actions, since G has to be realized by partial isometries.If an associative semimultiplicative set G has left cancellation, that is, for all s, t 1 , t 2 ∈ G, st 1 = st 2 implies t 1 = t 2 , then we are able to define a left reduced C * -algebra for G. Write (e g

An elementary Green imprimitivity theorem for inverse semigroups

“…We begin by recalling crossed products in the sense of Khoshkam and Skandalis [11] and Sieben [19], but use several notions from [6]. Let G denote an inverse semigroup.…”

Aspects of equivariant $KK$-theory in its generators and relations picture

“…The reference for group equivariant KK-theory is Kasparov [12], for groupoid equivariant KK-theory it is Le Gall [13], and for inverse semigroup equivariant KK-theory it is [3], or see [2] for a summary of the definitions. (We use the slightly adapted 'compatible' version of equivariant KK-theory as in [2].)…”

“…Skandalis' proof is non-equivariant, but it works also equivariant, see for example [3] for inverse semigroups G.…”

Inverse semigroup equivariant KK-theory and C^*-extensions

“…In this note, we shall however require none of them. We consider G-equivariant KK-theory as defined in [3] but make a slight adaption by making this theory compatible in the sense that we only allow for compatible Hilbert (bi)modules (cf. Definition 2.2).…”

“…There exists an associative Kasparov product in KK G as usual (see [3]). We list here some notions from [11].…”